If F is Lebesgue Integrable and Continuous at a Point C

Selected Topics of Real Analysis

Alexander S. Poznyak , in Advanced Mathematical Tools for Automatic Control Engineers: Deterministic Techniques, Volume 1, 2008

Theorem 16.25. (The Tonelli–Hobson theorem)

Assume that f is measurable on ${ℝ}^2$ and that at least one of two iterated integrals

$\begin{array}{ccc}{\displaystyle \underset{ℝ}{\int }\left[{\displaystyle \underset{ℝ}{\int }\left|f\left(x,y\right)\right|dx}\right]}dy& or& {\displaystyle \underset{ℝ}{\int }\left[{\displaystyle \underset{ℝ}{\int }\left|f\left(x,y\right)\right|dy}\right]}dy\end{array}$

exists. Then

(a)

$f\in ℒ\left(\mu \right)$ on ${ℝ}^2$ ;

(b)

The formula (16.116) holds.

Proof

Part (b) follows from (a) because of the Fubini's theorem (16.24). To prove (a) assume that the iterated integral

exists. Let $\left\{s_n\right\}$ be an increasing (nondecreasing) sequence of nonnegative step functions defined by the formula

$s_n\left(x,y\right)=\{\begin{array}{ccc}n& \text{if}& \left|x\right|\le n\text{ and}\left|y\right|\le n\\ 0& \text{if}& \text{otherwise}\end{array}$

Let also $f_n\left(x,y\right):=\min \left\{s_n\left(x,y\right),\left|f\left(x,y\right)\right|\right\}$ . Notice that both s_n and $\left|f\right|$ are measurable on ${ℝ}^2$ . So, f_n is measurable and, since

so f_n is dominated by a Lebesgue integrable function. Therefore, by Theorem 16.21 $f_n\in ℒ\left(\mu \right)$ on ${ℝ}^2$ . Hence, we can apply Fubini's theorem 16.24 to f_n along with the inequality

to obtain

${\displaystyle \underset{{ℝ}^2}{\int }{f}_{n}d\mu }={\displaystyle \underset{ℝ}{\int }\left[{\displaystyle \underset{ℝ}{\int }{f}_n\left(x,y\right)dx}\right]}dy\le {\displaystyle \underset{ℝ}{\int }\left[{\displaystyle \underset{ℝ}{\int }\left|f_n\left(x,y\right)\right|dx}\right]}dy$

Since $\left\{f_n\right\}$ is increasing this shows that $\underset{n\to \infty }{\lim }{\displaystyle {\int }_{{ℝ}^2}{f}_{n}d\mu }$ exists. But $\left\{f_n\left(x,y\right)\right\}\to \left\{\left|f_n\left(x,y\right)\right|\right\}$ almost everywhere on ${ℝ}^2$ . So, $\left|f\right|\in ℒ\left(\mu \right)$ on ${ℝ}^2$ . Since f is measurable, it follows that $f\in ℒ\left(\mu \right)$ on ${ℝ}^2$ which proves (a). The proof is similar if the other integral exists. Theorem is proven.

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Logarithmic Wavelets

Laurent Navarro , ... Michel Jourlin , in Advances in Imaging and Electron Physics, 2014

7 Main Notations

f

Finite energy function (signal or image)

$\overline{f}$

Complex conjugate of f

$\hat{f}$

Fourier transform of f

L ¹

Space of Lebesgue integrable functions

L ²

Hilbert space

G _f (v,b)

Short-time Fourier transform of f

F(u,v)

2-D Fourier transform of f

$G\left(x,y,{\mu }_0,v_0,x_0,y_0,\rho ,\theta ,\sigma ,\beta \right)$

2-D Gabor filter

CWT _f (a,b)

CWT of f

${\zeta }_{a,b}\left(x\right)$

Mother wavelet

a

Scaling factor

b

Translation factor

$CW{T}_f\left(\text{b,}\theta ,a\right)$

2-D CWT of f

${\left(V_j\right)}_{j\in \mathrm{\mathbb{Z}}}$

Subspaces in MRA analysis

${\phi }_{j,k}$

and ${\zeta }_{j,k}$ Scaling function and mother wavelet in MRA analysis

$\oplus$

Direct sum

$\otimes$

Tensor product

$h_{\zeta }$

Low-pass filter

$h_{\phi }$

High-pass filter

W _j

Wavelet space

$W{T}_{\phi }$

Approximation of wavelet coefficients

$W{T}_{\zeta }$

Detail of wavelet coefficients

$W{T}_{\zeta }^H$

2-D horizontal details of wavelet coefficients

$W{T}_{\zeta }^V$

2-D vertical details of wavelet coefficients

$W{T}_{\zeta }^{HV}$

2-D diagonal details of wavelet coefficients

T _f

Transmittance of f

$⨹$

LIP addition

$⨺$

LIP subtraction

$⨻$

LIP multiplication

$◬$

LIP product

$\mathrm{⊞}$

S-LIP addition

$\mathrm{⊠}$

S-LIP multiplication

$\mathrm{⊡}$

S-LIP product

$\psi$

Logarithmic generating function

${\psi }^{-1}$

Inverse logarithmic generating function

${\psi }_{S-LIP}$

S-LIP generating function

${\psi }_{S-LIP}^{-1}$

Inverse S-LIP generating function

${\zeta }_{\mathrm{▵}}\left(a,b\right)$

Logarithmic mother wavelet

$CW{T}_{\mathrm{▵}}\left(a,b\right)\left(f\right)$

Logarithmic CWT transform of f

${\otimes }_{\mathrm{▵}}$

Logarithmic tensor product

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Geometric Measure Theory in Banach Spaces

David Preiss , in Handbook of the Geometry of Banach Spaces, 2003

2.1 Differentiable measures

In much of modern analysis in finite-dimensional spaces, the role of pointwise derivative has been completely overshadowed by that of derivative in the sense of distributions. If ƒ:ℝ → ℝ is Lebesgue integrable, its distributional derivative may be defined as a Lebesgue integrable function g: ℝ → ℝ such that the formula for integration by parts

(1) ${\displaystyle \int {\varphi }^{\prime }\left(t\right)}ƒ\left(t\right)\text{d}t=-{\displaystyle \int \varphi \left(t\right)g\left(t\right)\text{d}t}$

holds for every smooth ϕ: ℝ → ℝ with bounded derivative. However, observing that in (1) the functions ƒ and g are only used as acting on functions by integration, i.e., as measures, we may consider it as defining that the distributional derivative of a signed measure μ is a signed measure v such that that ∫ϕ'(t) dμ(t) = – ∫(t) dν(t) for every smooth bounded ϕ: ℝ → ℝ with bounded derivative. On the real line, this generality is partly spurious, since it is easy to see that a measure μ on ℝ has this derivative if and only if it is a function of bounded variation. (Somewhat loosely, one says that the measure μ(E) = ∫ _E ƒ(t)dt is a function, namely, the function ƒ.) However, the derivative may well be a measure which is not a function, for example, the derivative of the function ƒ(t) = signum(t) is the Dirac measure. A similar approach is used in ℝ ⁿ to define distributional partial derivatives; and again their existence means that the measure is a function. In fact, it is again a function of bounded variation, usually by definition (see, for example, [61]).

The definition of distributional derivatives of measures admits a direct generalization to Banach spaces (where we have no notion of a measure being a function): the derivative of a (finite Borel) signed measure μ in direction w is a signed measure D_uμ such that

(2) ${\displaystyle \int D_w\varphi \left(x\right)}\text{d}\mu \left(x\right)=-{\displaystyle \int \varphi \left(x\right)\text{d}{D}_w\mu \left(x\right)}$

for every bounded continuously differentiable ϕ: X → ℝ with bounded derivative.

The definition immediately implies that the set of directions of differentiability of μ is a linear space, the mapping w → D_uμ is linear and that differentiation commutes with convolution, i.e., D_w (ν∗ μ) = ν ∗ D_uμ provided that D_uμ exists.

Directional derivatives of measures may be equivalenty defined by more direct formulae: derivative of μ in the direction w in Skorochod's sense is defined by (see [52, §21] for details)

(3) ${\displaystyle \int \varphi }\text{d}{D}_w\mu =-\underset{r\to 0}{\lim }{\displaystyle \int \frac{\varphi \left(x+rw\right)-\varphi \left(x\right)}{r}}\text{d}\mu \left(x\right)$

provided that the limit exists for every bounded continuous ϕ: X → R; the functional defined by the limit is necessarily an integral with respect to a measure. Another approach which was developed in finite-dimensional spaces by Tonneli needs essentially no modification in infinite-dimensional spaces: we require that μ has a disintegration

${\displaystyle \int \varphi }\text{d}\mu ={\displaystyle \int {}_Y}{\displaystyle \int {}_{ℝ}\varphi \left(y+tw\right){\psi }_y\left(t\right)}\text{d}t\text{d}\nu \left(y\right),$

where Y is a complement of span{w}, ν is a probability measure on Y and ψ are (right continuous) functions of bounded variation; under these conditions we define

(4) ${\displaystyle \int \varphi }\text{d}{D}_w\mu ={\displaystyle \int {}_Y}{\displaystyle \int {}_{ℝ}\varphi \left(y+tw\right)\text{d}{\psi }_y\left(t\right)}\text{d}\nu \left(y\right).$

It is easy to see that the derivatives of measures defined by (3) or (4) satisfy (2). If (2) holds, we obtain (3) by denoting ν_r (E) = λ{t ∈ [0, r]: tw ∈ E}/r, inferring from the formula for differentiation of convolution that

${\displaystyle \int \varphi }\text{d}{\nu }_r\ast D_w\mu =-{\displaystyle \int \frac{\varphi \left(x+rw\right)-\varphi \left(x\right)}{r}}\text{d}\mu \left(x\right)$

first for every bounded continuously differentiable ϕ: X → ℝ with bounded derivative and then, by approximation, for every bounded continuous ϕ: X → ℝ, and by letting r → 0. Finally, to obtain (4) from (2), we disintegrate

${\displaystyle \int h}\text{d}{D}_w\mu ={\displaystyle \int {}_Y}{\displaystyle \int {}_{ℝ}h\left(y+tw\right)\text{d}{\sigma }_y\left(t\right)}\text{d}\nu \left(y\right)$

and let ψ _y (t) = σ _y (−∞, t). By approximation, it suffices to show that (4) holds for every continuously differentiable function ϕ: X → ℝ with bounded derivative and with {t ∈ ℝ: ϕ(y + tw) ≠ 0 for some y ∈ Y} bounded. For any τ > 0 denote $g_{\tau }\left(y+tw\right)={\displaystyle {\int }_{-\infty }^t\varphi (y+sw)-\varphi (y+(s+\tau )w)\text{d}s}$ and use (2) and integration by parts to infer that

$\begin{array}{ll}{\int }^{}{D}_w{g}_{\tau }(x)\text{d}\mu (x)\hfill & =-{\int }^{}{g}_{\tau }(x)\text{d}{D}_w\mu (x)=-{{\int }^{}}_Y{{\int }^{}}_{ℝ}{g}_{\tau }(y+tw)\text{d}{\sigma }_y(t)\text{d}v(y)\hfill \\ \hfill & ={{\int }^{}}_Y{{\int }^{}}_{ℝ}{D}_w{g}_{\tau }(y+tw){\psi }_y(t)\text{d}t\text{d}v(y).\hfill \end{array}$

Since D_w g _τ (x) = ϕ(x) − ϕ(x + τw), (2) follows by letting τ → ∞.

Currently the most useful notion of derivative of a measure μ (often called differentiability in the sense of Fomin) is obtained by requiring additionally that D_wμ be absolutely continuous with respect to μ. This is equivalent to validity of (3) for every bounded Borel measurable function or to differentiability at t = 0 of the function assigning to t ∈ ℝ the measure μ shifted by tw when the space of measures is equipped with the usual norm. The Radon-Nikodým derivative of D_wμ with respect to μ is called the logarithmic derivative of μ in direction w; one readily sees that this term is justified in the finite-dimensional situation. All these notions have been treated as a special case of differentiability of mappings of the real line into the space of signed measures equipped with various topologies in [50]; another particular case of this treatment is the notion of differentiability of measures along vector-fields. (Of course, in this generality some of the equivalences mentioned above may fail.) Under very mild assumptions, these authors also prove the key formula $d{\mu }_a/d{\mu }_b=exp({\displaystyle {\int }_a^b{\varrho }_t(x)\text{d}t})$ , where ϱ _t is the logarithmic derivative of t ∈ ℝ → μ _t . (See [50] for the history of this formula and its applications.) In the setting when H is a sub-space of E consisting only of directions of logarithmic differentiability and h : X → H, one can, under appropriate assumptions, compute the logarithmic derivative of t → (id +th) _#μ and the Radon-Nikodým derivative d(id + th)_# μ/dμ from the derivative of h and directional logarithmic derivatives of μ – the latter gives a substitution theorem mentioned above (see [51]). The assumptions alluded to here are, of necessity, much stronger than those mentioned so far: since the formulas involve either the trace of the derivative of h in the direction of H or the determinant of id + th'(x). (This also explains why H is supposed to carry a Hilbert space structure.)

A Gaussian measure is logarithmically differentiable exactly in the directions of its Cameron–Martin space and the derivatives may be found explicitly. For these measures, the above results form just a beginning of the story; see, for example, [8] for much more.

The natural problem of unique determination of a measure by its logarithmic derivative has been answered negatively in [38]. (Prior to it, several authors noted that a positive answer would not only mean that some correspondence between functions and measures survives to the infinite-dimensional situation but would also have interesting applications.)

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Optimal Transportation

Y. Brenier , in Encyclopedia of Mathematical Physics, 2006

Monge's Optimal Transportation Problem

Theorem 2 is one of the numerous avatars of the so-called optimal transportation theory that goes back to Monge's mass transfer problem which addressed in 1781 the 'mémoire sur la théorie des déblais et des remblais' and was completely renewed by Kantorovich in the 1940s (see e.g., Rüschendorf and Rachev (1990) for instance). Let us quote a typical result, similar to Theorem 2, but without regularity assumptions on the data (see Brenier and Caffarelli (1992)):

Theorem 4

Let ρ ₀ be a non- negative Lebesgue integrable function on R ^d , such that

${\displaystyle {\int }_{{\mathit{R}}^d}{\rho }_0(x)\text{d}x=1}$

Then for any Borel probability measure ρ ₁(dy) with compact support on R ^d , there is a unique map T transporting ρ ₀(x)dx to ρ ₁(dy), which minimizes

${\displaystyle {\int }_{{\mathit{R}}^d}{|T(x)-x|}^2{\rho }_0(x)\text{d}x}$

where | · | denotes the Euclidean norm on R ^d . In addition, there is a Lipschitz continuous convex function Φ defined on R ^d such that T(x) = DΦ(x) for ρ ₀ almost every x ∈ R ^d , which implies:

${\displaystyle {\int }_{{\mathit{R}}^d}f(D\text{Φ}(x)){\rho }_0(x)\text{d}x={\displaystyle {\int }_{{\mathit{R}}^d}f(y){\rho }_1(\text{d}y)}}$

for all continuous functions f on R ^d .

Theorem 2, which can be interpreted as a regularity result with respect to Theorem 4, is the main output of Caffarelli's regularity theory for transportation maps with convex potentials (Caffarelli 1992). Caffarelli's analysis starts by a proof that Φ actually is a weak solution of the Monge–Ampère equation [3] in the sense of Alexandrov and is strictly convex. Then, Caffarelli shows that D ²Φ is Hölder continuous, as soon as ρ ₀ and ρ ₁ are Hölder continuous.

Notice that the convexity assumption for Ω₁ is crucial to insure the regularity of the convex potential. Caffarelli provided counter-examples when Ω₁ is made of two separate balls attached together by a sufficiently thin pipe.

Surprisingly enough, results such as Theorem 4 are related to concrete applications in, for example, astrophysics, image processing, etc. (Frisch et al. 2002, Haker and Tannenbaum 2003).

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Differentiation

Brian S. Thomson , in Handbook of Measure Theory, 2002

12.1 The cube basis

In the space ℝ ⁿ let α denote the collection of all pairs (I, ξ) where I is an n-dimensional cube and ξ ∈ I. The derivation basis D _o generated from α using the ordinary euclidean metric is called the "cube basis" and differentiation relative to this basis is usually called "ordinary differentiation".

Let λ _n denote Lebesgue measure on ℝ ⁿ . The basis cube basis D _o has the strong Vitali property relative to λ _n . There are a number of ways to prove this. The methods of Section 9 can be used, and indeed some of the details of the method are given already in that section. It follows then that the differentiation theorem holds for this. Namely for every locally Lebesgue integrable function f the set of points x ∈ ℝ ⁿ at which

(49) ${\mathcal{D}}_0\underset{I\to x}{\lim }\frac{1}{{\lambda }_n\left(I\right)}{\displaystyle {\int }_I\left|f\left(x\right)-f\left(y\right)\right|d}{\lambda }_n\left(y\right)=0.$

that is the Lebesgue set of f, includes λ _n -a.e. point of ℝ ⁿ . In particular, for λ _n -almost every point x ∈ ℝ ⁿ

(50) ${\mathcal{D}}_0\underset{I\to x}{\lim }\frac{1}{{\lambda }_n\left(I\right)}{\displaystyle {\int }_{I}f\left(y\right)d}{\lambda }_n\left(y\right)=f\left(x\right).$

Thus, in many ways, this seems to be the most natural and important derivation basis on ℝ ⁿ . A variant on this that would be as natural and have these same properties is to take the balls instead: Thus α consists of all pairs (I, ξ) where I = B(y, r) is a ball of radius r and x ∈ I (which requires that |x − y| < r). Note that in both cases the associated point ξ in the pairs (I, x) is in the set I but not required to be in any special geometric position in that set.

Consider, however, the question as to whether this same result holds true if λ _n is replaced by any locally finite Borel measure μ on ℝ ⁿ . An easy example shows that this is not the case. Let L be the line y = x in ℝ² and let μ(E) be the linear measure of E ∩ L. Then there is no strong Vitali property for D _o relative to μ and there is no differentiation theorem. An examination of this example shows that the position of the point ξ ∈ I for the pairs (I, ξ) seems to be a critical issue for the strong Vitali property.

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Ordinary Differential Equations

A. Rontó , M. Rontó , in Handbook of Differential Equations: Ordinary Differential Equations, 2008

1 Notation

(1)

ℝ   =   (–∞,∞), ℝ₊  =   [0, ∞), ℤ   =   {0, ±   1, ±   2,…}, ℕ   =   {1, 2,…}.

(2)

C _T ⁿ is the Banach space of continuous vector-functions x  =   (x _k ) _{k  =   1} ⁿ   :   (−∞,   ∞)   →   ℝ ⁿ periodic with period T (i.e., x(t)   = x(t  + T) for any t ∈ (–∞,∞)), with the usual norm

(1.1) $\mathit{x}\mapsto \underset{\mathit{k}=1,2,\dots ,\mathit{n}}{\text{max}}\underset{\mathit{t}\in \left[0,\mathit{T}\right]}{\text{max}}\left|{\mathit{x}}_{\mathit{k}}\left(\mathit{t}\right)\right|.$

(3)

C([0,T]   × D, ℝ ⁿ ), where D  ⊂   ℝ ⁿ is a compact set, stands for the Banach space of continuous vector-functions x  =   (x _k ) _{k  =   1} ⁿ   :   [0, T]   →   ℝ ⁿ with norm (1.1).

(4)

L ₁([0,T], ℝ ⁿ ) is the Banach space of the Lebesgue integrable functions x  =   (x _k ) _{k  =   1} ⁿ   :   (−∞,   ∞)   →   ℝ ⁿ with the norm

$\mathit{x}\mapsto \underset{\mathit{k}=1,2,\dots ,\mathit{n}}{\text{max}}{\int }_0^{\mathit{T}}\left|{\mathit{x}}_{\mathit{k}}\left(\mathit{t}\right)\right|\mathrm{d}\mathit{t}.$

(5)

GL _n (ℝ) is the algebra of square real matrices of dimension n.

(6)

1 _n is the n-dimensional unit matrix.

(7)

The symbol r(A) stands for the spectral radius of a square matrix A.

(8)

I is the identity operator in various spaces.

(9)

∂Ω is the boundary of a set Ω   ⊂   ℝ^k.

(10)

If Ω₁  ⊂   ℝ ⁿ , Ω₂  ⊂   ℝ ⁿ , and {α ₁,α ₂}   ⊂   ℝ, then α ₁Ω₁  + α ₂Ω₂ := {α ₁ x ₁  + α ₂ x ₂ | x1 ∈ Ω₁, x ₂ ∈ Ω₂}.

(11)

For a point x   =   col(x ₁,x ₂,…,x_n ) ∈ ℝ ⁿ , we denote by |x| the vector |x|   =   col(|x ₁|, |x ₂|,…, |x_n |). Similarly, the signs |   ·   |, max, min, and   ≤   for vectors are understood componentwise.

(12)

Let β  =   col(β ₁,…,β_n ) be a vector from ℝ₊ ⁿ . By the β-neighbourhood of a point y ∈ ℝ ⁿ we understand the set B(y,β) of points defined as follows:

(1.2) $\mathit{B}\left(\mathit{y},\mathit{\beta }\right):=\left\{\mathit{x}\in {\mathrm{ℝ}}^{\mathit{n}}|\left|\mathit{x}-\mathit{y}\right|\le \mathit{\beta }\right\}.$

(13)

For any vector β  ∈   ℝ₊ ⁿ , by D_β , we mean the set

(1.3) ${\mathit{D}}_{\mathit{\beta }}:=\left\{\mathit{x}\in \mathit{D}|\mathit{B}\left(\mathit{x},\mathit{\beta }\right)\subset \mathit{D}\right\}$

that consists of points x ∈ ℝ ⁿ lying in D together with their β-neighbourhoods. The symbol □ denotes the end of a proof.

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The Lebesgue Integral

Don Hong , ... Robert Gardner , in Real Analysis with an Introduction to Wavelets and Applications, 2005

Exercises

1.

Let 〈f _n 〉 be a sequence of measurable functions on E. If (i) there exists an integrable function F on E such that |f _n | ≤ F a.e. on E for $n\in \mathbb{N}$ , and (ii) f _n → f a.e. on E, then prove $f_n\stackrel{m}{⟶}f$ as n → ∞.

2.

Suppose 〈g _n 〉 is a sequence of integrable functions on E and g _n → g a.e. with g integrable. If the sequence 〈f _n 〉 of measurable functions satisfies:

(i)

|f _n | ≤ g _n for $n\in \mathbb{N}$

(ii)

f _n → f a.e. on E,

then prove ${\text{lim}}_{n\to \infty }{\int }_E{f}_{n}dx={\int }_{E}fdx$

3.

Let L[a, b ] denote the class of all Lebesgue integrable functions on [ a, b]. Show that for f ∈ L[a, b] and for all ∈ > 0, there is

(i)

a bounded measurable function g such that ${\int }_a^b|f(x)-g(x)|dx<\varepsilon$ ;

(ii)

a continuous function h such that ${\int }_a^b|f(x)-h(x)|dx<\in$

(iii)

a polynomial function p such that ${\int }_a^b|f(x)-p(x)|dx<\in$

(iv)

a step function s such that ${\int }_a^b|f(x)-s(x)|dx<\in$

4.

If f ∈ L[a, b], then prove ${\text{lim}}_{n\to \infty }{\int }_a^{b}f(x)\text{cos}nxdx=0$ and ${\text{lim}}_{n\to \infty }{\int }_a^{b}f(x)\text{sin}nxdx=0$ . [Hint: Consider a step function f first.]

5.

Show by example that the inequality in Fatou's Lemma is not in general an equality even if the sequence of functions 〈f _n 〉 converges everywhere.

6.

If f ∈ L[a, b], then prove ${\text{lim}}_{n\to \infty }{\int }_a^{b}f(x)\left|\text{cos}nx\right|dx=\frac{2}{\pi }{\int }_a^{b}f(x)dx$ .

7.

If f ∈ L[a, b] and for $k\in N,{\int }_a^b{x}^{k}f(x)dx=0,$ , then prove f = 0 a.e. on [a, b].

8.

If $f\in L(ℝ)$ and for any compact supported continuous function g, ${\int }_{ℝ}f(x)g(x)dx=0$ , then prove f = 0 a.e.

9.

If 〈f _n 〉 is a sequence of integrable functions on E satisfying $f_n\stackrel{m}{⟶}f$ , then prove ${\int }_{E}f(x)dx\le {\underset{—}{lim}}_{n\to \infty }{\int }_E{f}_{n}dx$ .

10.

Let f ∈ L[0, 1]. Then x ⁿ f(x) ∈ L[0, 1] for any $n\in \mathbb{N}$ and ${\text{lim}}_{n\to \infty }{\int }_0^1{x}^{n}f(x)dx=0$ .

11.

If f _n , f ∈ L(E) and ${\int }_E|f_n-f|dx\to 0$ , then $f_n\stackrel{m}{⟶}f$ .

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The Henstock-Kurzweil Integral

Benedetto Bongiorno , in Handbook of Measure Theory, 2002

3 Further Riemann-type integrals on the real line

The method of Henstock and Kurzweil was adapted in many ways to different situations. The first and most striking modification was done by McShane (1969). It concerns the notion of partition. A collection $P={\left\{\left(A_i,x_i\right)\right\}}_{i=1}^P$ of pairwise disjoint intervals A_n and points $x_i\in \left[a,b\right]$ is said to be a McShane partition if $\left[a,b\right]={\cup }_i{A}_i$ Remark that each partition in the sense of Section 1 is a McShane partition, but not vice versa.

3.1 The McShane integml

DEFINITION 3.1

It is said that $f:\left[a,b\right]\to ℝ$ is McShane integrable on [a. b] whenever there is a real number I satisfying the following condition: for each $\epsilon >0$ there exists a gauge δ such that

for each δ-fine McShane partition P of [a.b].

The fact that in a McShane partition $\left\{\left(A_i,x_i\right)\right\}$ the point JIT, may be chosen outside the interval A_i increases the number of δ-fine partitions, so it is more difficult for a function to be integrable. In fact,

THEOREM 3.2 (McShane (1969))

A function f is McShane integrable on [a.b] if and only if it is Lebesgue integrable on [a, b].

3.2 The C-integral

Bruckner et al. (1986) remarked that the solution of the problem of primitives provided by Denjoy, Perron, Kurzweil and Henstock possesses a generality which is not needed for this purpose. In fact the function

$F\left(x\right)=\{\begin{array}{cc}x\sin \frac{1}{x^2},& 0<x⩽1,\\ 0,& x=0.\end{array}$

is a primitive for the Denjoy-Perron-Kurzweil-Henstock integral, but it is neither a Lebesgue primitive, nor a differentiable function, or a sum of a Lebesgue primitive and a differentiable function.

A Riemann-type definition for the minimal integral which includes Lebesgue integrable functions and derivatives was given by Bongiorno (1996) and Bongiomo et al. (2000), using the following small modification of the McShane integral.

Definition 3.3

A function $f:\left[a,b\right]\to ℝ$ it is said to be C-integrable on [a, b] if there exists a constant I such that for each $\epsilon >0$ there is a gauge δ so that

for each δ-fine McShane partition $\left\{\left(A_1,x_1\right)\dots \mathrm{.}\left(A_P,x_P\right)\right\}$ satisfying the condition ${\displaystyle {\sum }_{i=1}^P\text{dist}\left(x_i,A_i\right)<1/\epsilon }$ .

The mentioned minimality of this integral is an immediate consequence of the next theorem.

THEOREM 3.4 (Bongiorno et al. (2000))

A function $f:\left[a,b\right]\to ℝ$ is C-integrable if and only if there exist a derivative f ₁ and a Lebesgue integrable function f ₂ such that $f=f_1+f_2$ .

3.3 Integmls induced by differentiation bases

We recall that a differentiation basis on [a, b] is a collection $\Delta$ of pairs (A, x), where A is a subinterval of [a, b] and x is a point of A, such that $\inf \left\{\lambda \left(A\right):\left(A,x\right)\in \Delta \right\}=0$ for each $x\in \left[a,b\right]$ . A partition $P={\left\{\left(A_i,x_i\right)\right\}}_{i=1}^p$ is called a $\Delta$ -partition whenever $\left(A_i,x_i\right)\in \Delta$ for $i=1,2,\dots ,p$ .

Let $ℬ$ be a family of differentiation bases $\left\{\Delta \left(\alpha \right):\alpha \in ℐ\right\}$ satisfying the partitioning property; i.e., a $\Delta \left(\alpha \right)$ -partition of [a, b] exists for each $\alpha \in ℐ$ .

A function $f:\left[a,b\right]\to ℝ$ is said to be $ℬ$ -integrable, with integral I, on [a.b] if for every $\epsilon >0$ there exists a differentiation basis $\Delta \left(\alpha \right)\in ℬ$ such that $\left|{\displaystyle {\sum }_{P}f-I}\right|<\epsilon$ . for each $\Delta \left(\alpha \right)$ -partition P of [a, b]. The number I is called the $ℬ$ -integral of f on [a, b].

Examples of such integrals are the dyadic integral, the approximate integral, the symmetric integral and the approximate symmetric integral.

3.3.1 The dyadic integral

Let a and b be dyadic numbers and let δ be a gauge on [a, b]. We denote by ${\Delta }_d\left(\delta \right)$ the class of all pairs (A,x) such that $x\in A\subset \left[a,b\right],A=\left[j/2^k,\left(j+1\right)/2^k\right]$ , for some naturals j and k. and $A\subset \left]x-\delta \left(x\right),x+\delta \left(x\right)\right[$ . Let ${ℬ}_d=\left\{{\Delta }_d\left(\delta \right)\right\}$ , where δ runs on the family of all gauges on [a, b]. The ${ℬ}_d$ -integral is called the dyadic integral. It was studied in many papers, in particular in connection with some problem in Dyadic Harmonic Analysis (see Skvortsov (1969, 1988, 1992-93), Kahane (1988) and Gordon (1990-91b)).

The primitives have been characterized by Bongiorno et al. (2000) by an extension of Theorem 2.5 in which the variational measure is constructed using ${ℬ}_d$ partitions.

3.3.2 The approximate integral

Let $\left\{S_x:x\in \left[a,b\right]\right\}$ be a family of measurable subsets of [a, b] such that $x\in S_x$ and x is a point of density of S_x . The class of all pairs $\left(\left[c,d\right],x\right)$ such that $c,d\in S_x$ and $x\in \left[c,d\right]$ is, by definition, an approximate differentiation basis.

Let ${ℬ}_{\text{ap}}$ be the family of all approximate differentiation bases. The ${ℬ}_{\text{ap}}$ -integral is called the approximate integral.

A Perron equivalent definition of the approximate integral was defined by Burkill (1931). The approximate integral was studied by Henstock (1991), Carrington and Pacquement (1972), Bullen (1983), Liao (1987), Lee (1989), Gordon (1990-91a), Skvortsov (1992-93), Fu (1993-94), Liao and Chew (1993-94), Liao (1993-94), Gordon (1994), Lu (1996-97), Bongiomo et al. (2000).

The problem of recovering an approximately continuous function from its approximate derivative is solved by the approximate integral (see Gordon (1994)), while the Denjoy-Khintchine integral recovers the continuous primitives only (see Saks (1964) and Khintchine(1927)).

A function F is the primitives of f with respect the approximate integral if and only if it is approximately differentiable on [a, b] except for a countable set with approximate derivative $F_{{}^{\text{ap}}}^{\text{'}}=f$ almost everywhere on [a. b]. A different characterization in terms of generalized absolutely continuous function was given by Gordon (1994-95):

THEOREM 3.5

A function $F:\left[a,b\right]\to ℝ$ is a primitive of the approximate integral if and only if it is $AC{G}_{\Delta }$ ; i-e., there is a sequence of measurable sets $\left\{E_n\right\}$ such that $\left[a,b\right]={\cup }_n{E}_n$ , and for each $\epsilon >0$ and each $n\in \mathbb{N}$ there exist $\eta >0$ and an approximate differentiable base ${\Delta }_n$ such that

for each ${\Delta }_n$ -partition $\left\{\left(\left]c_i,d_i\right[,x_i\right):i=1,2,\dots ,p\right\}$ with ${\displaystyle {\sum }_{x_i\in E_n}\left(d_i-c_i\right)<\eta }$ .

Convergence theorems are due to Darmawijaya and Lee (1988).

3.3.3 The symmetric integral

Given a gauge δ on [a.b]. we denote by ${\Delta }_{\text{sym}}\left(\delta \right)$ the collection of all pairs (I, x) such that $I=\left[x-h,x+h\right]$ , with $0<h<\delta \left(x\right)$ and $\begin{array}{cc}a<x-h,& x+h<b\end{array}$ , or $I=\left[a,a+h\right]$ with $0<h<\delta \left(x\right)$ , or $I=\left[b-h,b\right]$ with $0<h<\delta \left(x\right)$ . Let ${ℬ}_{\text{sym}}=\left\{{\Delta }_{\text{sym}}\left(\delta \right)\right\}$ , where δ runs on the family of all gauges on [a. b]. The ${ℬ}_{\text{sym}}$ -integral is usually called the symmetric integral.

The symmetric integral was studied by Henstock (1963), Kurzweil and Jarník (1987, 1990a) and Thomson (1994). In fact the integral considered by Thomson (called the $\left(R_S^I\right)$ -integral) is somewhat more general than the ${\Delta }_{\text{sym}}$ -integral. It is defined by special partitions called symmetric partitions. For each gauge δ, a ${\Delta }_{\text{sym}}$ is said to be a symmetric partition if the conditions $\begin{array}{cc}a\in A_i,& b\in A_j\end{array}$ imply $\text{λ}\left(A_i\right)=\text{λ}\left(A_j\right).$

Remark that ${\Delta }_{\text{sym}}$ -(respectively $\left(R_S^I\right)$ -) integrability on [a,b] does not imply ${\Delta }_{\text{sym}}$ -(respectively $\left(R_s^{\text{l}}\right)\text{-})$ -integrability on each subinterval of [a, b] and that ${\Delta }_{\text{sym}}$ -(respectively $\left(R_S^l\right)$ -) integrability on contiguous intervals does not imply ${\Delta }_{\text{sym}}$ -(respectively $\left(R_S^l\right)$ integrability on their union.

The primitives of the $\left(R_S^l\right)$ -integral are characterized by symmetrically ACG ^* functions defined on all [a, b] except possibly some countable set (see Thomson (1994, Theorem 9.34)), or, in some particular cases, by an extension of Theorem 2.5 in which the variational measure is constructed using ${ℬ}_{\text{sym}}$ partitions (see Bongiorno et al. (2000, Corollary 5)).

Remark also that in some very simple situations the symmetric integral ${\Delta }_{\text{sym}}$ (respectively $\left(R_s^l\right)$ ) may be used to solve the coefficient problem for trigonometric series; i.e., if

$\begin{array}{llll}f\left(x\right)=\frac{a_0}{2}+{\displaystyle \sum _{k=0}^{\infty }\left(a_k\cos kx+b_k\sin kx\right)}.\hfill & for\hfill & each\hfill & x\in ℝ.\hfill \end{array}$

how may the coefficients of the series be determined!

Of course if f is Lebesgue integrable then the coefficients are determined by the usual Fourier formulas using Lebesgue integrals. But in general f is not Lebesgue integrable, neither Henstock-Kurzweil integrable. This is for example the case of each function $f\left(x\right)={\displaystyle {\sum }_n{b}_n\sin nx}$ with $b_n>0$ and ${\displaystyle {\sum }_n{b}_n/n=+\infty }$ . Nevertheless, even in the most general case $b_n\searrow 0$ , f is symmetrically integrable (in both senses) on any interval of length $2\pi$ and the coefficients are determined by the usual Fourier formulas using the symmetric integrals.

3.3.4 The approximate symmetric integral

A complete solution for the coefficient problem is given by the approximate symmetric integral, introduced by Preiss and Thomson (1989).

A differentiation basis $\Delta$ is said to be a measurable approximate symmetric differentiation basis if there is a measurable set $T\subset ℝ\times \left]0,\infty \right[$ such that

(i)

$\left(\left[x-t,x+t\right]\right)x)\in \Delta$ whenever $\left(x,t\right)\in T$ ,

(ii)

$\lim {\sup }_{h\searrow 0}\left|\left\{t\in \left(0,h\right):\left(x,t\right)\notin T\right\}\right|/h=0$ , for each $x\in ℝ$ .

Let ${ℬ}_{\text{as}}$ be the family of all measurable approximate symmetric differentiation bases. As before, in the definition of the ${ℬ}_{\text{as}}$ -integral the partitions are "symmetric partitions". The resulting integral is the approximate symmetric integral.

Solutions of the coefficients problem have been also done by Denjoy (1941-49), Marcinkiewicz and Zygmund (1936), James (1950) and Burkill (1951). They involve the inversion of a second order symmetric derivative:

$S{D}_{2}F\left(x\right)=\underset{h\to 0+}{\lim }\frac{F\left(x+h\right)+F\left(x-h\right)-2F\left(x\right)}{h^2}.$

A Riemann-type integral which gives F in function of SD₂ F was defined by Freiling et al. (1997). The definition of this integral involves several elaborate partitioning arguments for rectangles in the plane.

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Periodic Functions Generated as Solutions of Nonlinear Differential-Difference Equations1

G. STEPHEN JONES , in International Symposium on Nonlinear Differential Equations and Nonlinear Mechanics, 1963

1 Introduction

Consider functional equations of the form

(1) $f^{\prime }\left(x\right)=-\alpha {\displaystyle \sum _{i=1}^n{\lambda }_i}f\left(x-{\tau }_i\right)\left[af{\left(x\right)}^2+bf\left(x\right)+1\right],$

where 0 < τ₁ < τ₂ < … < τ _n and α, a, b and the λ _i 's are real parameters. Such equations, usually categorized as differential-difference equations with retarded arguments, occur in an impressively wide and varied field of applications. Most prevalent among these are control systems, biological growth behavior, and econometrics. One of the simplest members of this class,

(2) $f^{\prime }\left(x\right)=-\alpha f\left(x-1\right)\left[1+f\left(x\right)\right],$

even appears in the application of probability methods in the determination of the asymptotic density of prime numbers. For further discussion of these applications the reader is referred to Bellman [ 1 ], Lord Cherwell [ 2 ], Cunningham [ 3 ], and Wright [ 11 ].

Now in defining what we shall mean by a solution of equation (1) , we specify φ to be a bounded and Lebesgue integrable function defined on the interval (— τ _n , 0]. A function f such that f(x) = φ(x) for x in (— τ _n , 0] and satisfying the equation

(3) $f\left(x\right)=f\left(0\right)-\alpha {\displaystyle \sum _{i=1}^n{\lambda }_i}{\displaystyle {\int }_0^{x}f\left(t-{\tau }_i\right)}\left[af{\left(t\right)}^2+bf\left(t\right)+1\right]dt,$

for x in some interval (0, X), X > 0, is referred to as a solution of equation (1) on (0, X) corresponding to the initial function φ. Using the method of successive approximation it is easily shown that corresponding to each such initial specification φ, there exists an interval (0, X) on which equation (1) has a unique solution. It follows that a solution is uniquely determined by its specification on any interval of length τ _n within its domain of definition.

In the sequel φ always denotes an initial function of the type described in the previous paragraph and f denotes the corresponding solution of equation (1). The notation φ ∼ f signifies that f is the solution of (1) corresponding to φ. In addition, we assume throughout that α > 0 and that b ² — 4a ≧ 0 which implies the quadratic equation

(4) $a{y}^2+by+1=0$

has real roots. For a ≠ 0 we denote these roots by r ₁ and r ₂ where | r ₁ | < | r ₂ |. If a = 0 then the single root of (4) is denoted r ₁. Since we can do so without loss of generality, we specify r ₁ to be negative.

Now if we suppose that r ₁ ≠ r ₂ and choose an initial function φ such that φ(0) ≠ r _j , j = 1, 2, then by continuity there exists a solution f on an interval (0, X] for which f(x) ≠ r _j for all x in (0, X]. Equation (1) can be rewritten

$\frac{f^{\prime }\left(x\right)}{\left(f\left(x\right)-r_1\right)\left(f\left(x\right)-r_2\right)}=-\alpha a{\displaystyle \sum _{i=1}^n{\lambda }_i}f\left(x-{\tau }_i\right)$

for x in (0, X], since f(x) ≠ r _j , and through obvious manipulations we get

(5) $\frac{f\left(x\right)-r_1}{f\left(x\right)-r_2}=\frac{\phi \left(0\right)-r_1}{\phi \left(0\right)-r_2}\exp \left[\alpha a\left(r_2-r_1\right){\displaystyle \sum _{i=1}^n{\lambda }_i}{\displaystyle {\int }_0^{x}f\left(t-{\tau }_i\right)dt}\right],$

and

(6) $f\left(x\right)=\frac{r_1\left(\phi \left(0\right)-r_2\right)-r_2\left(\phi \left(0\right)-r_1\right)\exp \left[\alpha a\left(r_2-r_1\right){\displaystyle \sum _{i=1}^n{\lambda }_i}{\displaystyle {\int }_0^{x}f\left(t-{\tau }_i\right)dt}\right]}{\left(\phi \left(0\right)-r_2\right)-\left(\phi \left(0\right)-r_1\right)\exp \left[\alpha a\left(r_2-r_1\right){\displaystyle \sum _{i=1}^n{\lambda }_i}{\displaystyle {\int }_0^{x}f\left(t-{\tau }_i\right)dt}\right]}.$

In a similar fashion when r ₁ = r ₂ or a = 0 we get

(7) $f\left(x\right)=r_1+\left(\phi \left(0\right)-r_1\right){\left[1+\alpha a\left(\phi \left(0\right)-r_1\right){\displaystyle \sum _{i=1}^n{\lambda }_i}{\displaystyle {\int }_0^{x}f\left(t-{\tau }_i\right)dt}\right]}^{-1}$

or

(8) $f\left(x\right)=r_1+\left(\phi \left(0\right)-r_1\right)\exp \left[-\alpha b{\displaystyle \sum _{i=1}^n{\lambda }_i{\displaystyle {\int }_0^{x}f\left(t-{\tau }_i\right)dt}}\right]$

respectively for x in some interval (0, X].

It is our purpose to discuss briefly conditions under which equation (1) can be shown to have non-constant periodic solutions and to indicate a method of proof. Also, we shall describe some of the general properties of these solutions and present three illustrative examples which essentially characterize the behavior of the whole class.

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Generalizations of Riemann Integration

Agamirza Bashirov , in Mathematical Analysis Fundamentals, 2014

10.6* Lebesgue Integral

In this section we briefly outline measure and Lebesgue integration theory on subsets of $\mathbb{R}$ . For simplicity, we restrict ourself to a bounded closed interval $[a\text{,}b]$ .

At first, note that the Kurzweil–Henstock integration does not preserve some useful properties of the Riemann integration. This is the price, paid for the wideness of $\mathit{KH}(a\text{,}b)$ . One of such properties was noticed in Example 10.38. Another example is as follows.

Example 10.39

$f\in \mathit{KH}(a\text{,}b)$ does not necessarily imply $\mid f\mid \in \mathit{KH}(a\text{,}b)$ unlike Corollary 9.21(a) of the Riemann integration. Indeed, consider the function $f$ from Example 10.38. It was shown that $f\in \mathit{KH}(a\text{,}b)$ . We also have

${\int }_{1/n}^1\mid f(x)\mid \mathit{dx}={\displaystyle \sum _{k=1}^{n-1}}(k+1)\left(\frac{1}{k}-\frac{1}{k+1}\right)={\displaystyle \sum _{k=1}^{n-1}}\frac{1}{k}\text{.}$

Since the harmonic series diverges, by Theorem 10.32 we obtain $\mid f\mid \notin \mathit{KH}(0\text{,}1)$ .

Definition 10.40

If $f\in \mathit{KH}(a\text{,}b)$ is so that $\mid f\mid \in \mathit{KH}(a\text{,}b)$ , then $f$ is said to be integrable in the Lebesgue sense. In this case, the Kurzweil–Henstock integral of $f$ is also called the Lebesgue integral of $f$ . The collection of all Lebesgue integrable functions on $[a\text{,}b]$ is denoted by $L(a\text{,}b)$ .

This is a very unusual definition of the Lebesgue integral. Ordinarily, the Lebesgue integral is defined by partitioning the range of the integrands and forming respective integral sums. This requires extension of the concept of length from intervals to other subsets, to as many of them as possible, because the inverse image of an interval fails to be an interval in general. Such an extension is called a Lebesgue measure. The subsets to which the Lebesgue measure can be applied are said to be Lebesgue measurable. This gave raise to measure theory, where besides the Lebesgue measure (a natural length of intervals), other measures are considered as well. Each such measure $\lambda$ is related to some increasing function $f$ by letting the measure of the interval $[\alpha \text{,}\beta ]$ to be

$\lambda ([\alpha \text{,}\beta ])=f(\beta )-f(\alpha )\text{.}$

Generalizing this to functions $f$ from $\mathit{BV}(a\text{,}b)$ , the measures with negative values are defined as well, which are called charges.

Measure theory has many implications, justifying its significance. For example, in 1933 Kolmogorov ⁸² founded modern probability theory, interpreting a probability as a specific measure. Another implication goes to the foundations of mathematics. In 1964 Solovay ⁸³ showed that if the axiom of choice is made stronger by considering only countable families of nonempty sets, then all subsets of $\mathbb{R}$ are Lebesgue measurable. Earlier, an example of a nonmeasurable set in the Lebesgue sense was constructed on the base of the axiom of choice by Vitali. ⁸⁴

The relationship between the functions integrable in the Riemann, Lebesgue, and Kurzweil–Henstock senses can be expressed by

$R(a\text{,}b)\subset L(a\text{,}b)\subset \mathit{KH}(a\text{,}b)\text{,}$

stressing on the fact that the subset relations are strict. Indeed, $R(a\text{,}b)\subseteq L(a\text{,}b)$ since $R(a\text{,}b)\subset \mathit{KH}(a\text{,}b)$ and $f\in R(a\text{,}b)$ implies $\mid f\mid \in R(a\text{,}b)$ (Corollary 9.21(a)). Also, $R(a\text{,}b)\ne L(a\text{,}b)$ since Dirichlet's 1st function $f$ from Example 9.4 does not belong to $R(a\text{,}b)$ , while $\mid f\mid =f\in \mathit{KH}(a\text{,}b)$ (Example 10.30) and, therefore, $f\in L(a\text{,}b)$ . Furthermore, $L(a\text{,}b)\ne \mathit{KH}(a\text{,}b)$ by Example 10.39, while $L(a\text{,}b)\subseteq \mathit{KH}(a\text{,}b)$ by Definition 10.40.

$L(a\text{,}b)$ is a significant class of functions between $R(a\text{,}b)$ and $\mathit{KH}(a\text{,}b)$ . Unlike $R(a\text{,}b)$ and $\mathit{KH}(a\text{,}b)$ , it is possible to define a powerful norm in $L(a\text{,}b)$ . For this, we define an equivalence relation in $L(a\text{,}b)$ by letting $f\sim g$ if $f(x)=g(x)$ almost everywhere on $[a\text{,}b]$ . Denote the respective quotient set again by $L(a\text{,}b)$ : if $f\in L(a\text{,}b)$ , then $f$ is interpreted as a representative of all functions from $L(a\text{,}b)$ , which are equal to $f$ almost everywhere on $[a\text{,}b]$ . For $p\ge 1$ , we denote by $L_p(a\text{,}b)$ the collection of all $f\in L(a\text{,}b)$ , satisfying

${\int }_a^b\mid f(x){\mid }^p\mathit{dx}<\infty \text{.}$

Clearly, $L_1(a\text{,}b)=L(a\text{,}b)$ . For $f\in L_p(a\text{,}b)$ , we define

$\parallel f\parallel ={\left({\int }_a^b\mid f(x){\mid }^p\mathit{dx}\right)}^{1/p}\text{.}$

With this norm, $L_p(a\text{,}b)\text{,}p\ge 1$ , are useful Banach spaces. They reach most useful properties at $p=2$ .

Lebesgue integration removes the drawback of the Kurzweil–Henstock integration discussed in Example 10.38: if $f\in L(a\text{,}b)$ , then $F\in C(a\text{,}b)\cap \mathit{BV}(a\text{,}b)$ , where $F$ is a function defined by Eq. (9.10). Indeed, for every partition $\mathcal{P}=\{x_0\text{,}\dots \text{,}{x}_n\}$ of $[a\text{,}b]$ , we have

${\displaystyle \sum _{i=1}^n}\mid F(x_i)-F(x_{i-1})\mid \le {\displaystyle \sum _{i=1}^n}{\int }_{x_{i-1}}^{x_i}\mid f(x)\mid \mathit{dx}={\int }_a^b\mid f(x)\mid \mathit{dx}<\infty \text{,}$

implying $V(F\text{;}a\text{,}b)<\infty$ . This suggests the following very important subclass of $C(a\text{,}b)\cap \mathit{BV}(a\text{,}b)$ .

Definition 10.41

A function $f\in C(a\text{,}b)\cap \mathit{BV}(a\text{,}b)$ is said to be absolutely continuous if it can be represented in the form

$f(x)=f(a)+{\int }_a^{x}g(t)\mathit{dt}\text{,}a\le x\le b\text{,}$

for some $g\in L(a\text{,}b)$ . The collection of all absolutely continuous functions on $[a\text{,}b]$ is denoted by $\mathit{AC}(a\text{,}b)$ .

In Definition 10.41, $g$ and $f^{\prime }$ can be identified since the elements of $L(a\text{,}b)$ are classes of almost everywhere equal functions on $[a\text{,}b]$ and $g=f^{\prime }$ almost everywhere on $[a\text{,}b]$ . The class $\mathit{AC}(a\text{,}b)$ , endowed with the norm

$\parallel f\parallel =\mid f(a)\mid +{\int }_a^b\mid f^{\prime }(x)\mid \mathit{dx}\text{,}$

becomes a Banach space. Notice that the subset relation

$\mathit{AC}(a\text{,}b)\subset C(a\text{,}b)\cap \mathit{BV}(a\text{,}b)$

is strict because the Cantor function $f$ from Section 8.2 is continuous and monotone on $[0\text{,}1]$ and, therefore, belongs to $C(a\text{,}b)\cap \mathit{BV}(a\text{,}b)$ , but $f\notin \mathit{AC}(0\text{,}1)$ by Example 10.37. Regarding this case, it is useful to mention the following result as well, the proof of which is based on the Vitali covering theorem, an advanced theorem of measure theory.

Theorem 10.42

Every monotone function $f:[a\text{,}b]\to \mathbb{R}$ is differentiable almost everywhere on $[a\text{,}b]$ .

Thus, the discontinuity points of a monotone function form a countable set (Corollary 8.4), whereas the set of points, at which it is nondifferentiable, has measure zero. Both these sets are regarded as "negligible sets," but in different senses. Theorem 10.42 automatically extends to functions of bounded variation and suggests the following.

Definition 10.43

A function $f\in C(a\text{,}b)\cap \mathit{BV}(a\text{,}b)$ is said to be singular if it is nonconstant and $f^{\prime }(t)=0$ almost everywhere on $[a\text{,}b]$ .

Example 10.37 tells us that the Cantor function is a singular function. Moreover, every continuous function of bounded variation can be uniquely decomposed to the sum of absolutely continuous and singular components. Then by Theorem 8.26, every $f\in \mathit{BV}(a\text{,}b)$ can be composed into the sum of three components

$f=f_{\text{jump}}+f_{\text{ac}}+f_{\text{sing}}\text{,}$

where $f_{\text{jump}}\text{,}{f}_{\text{ac}}$ , and $f_{\text{sing}}$ are the jump, absolutely continuous, and singular components with $f_{\text{cont}}=f_{\text{ac}}+f_{\text{sing}}$ . Respectively, every measure on $\mathbb{R}$ is composed by jump, absolutely continuous, and singular components. In 1913 Radon ⁸⁵ modified absolutely continuous measures to ${\mathbb{R}}^k$ , and in 1930 Nikodym ⁸⁶ extended it to abstract measures.

Concluding this chapter, where we left far from the framework of classical mathematical analysis, we present the diagram in Figure 10.1, showing the relationship of basic classes of functions.

Figure 10.1. Relationship of basic classes of functions.

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