Lebesgue integral

Top Riemann integration, Bottom Lebesgue

Lebesgue integral is a generalization of the Riemann integral to a wider class of functions .

All functions defined on a finite segment of the number line and Riemann integrable are also Lebesgue integrable, and in this case both integrals are equal. However, there is a large class of functions defined on a segment and Lebesgue integrable, but non-integrable by Riemann. Also, the Lebesgue integral can make sense for functions defined on arbitrary sets ( Frechet integral ).

The idea of constructing the Lebesgue integral ^[1] is that instead of dividing the domain of definition of the integrand into parts and then drawing up the integral sum of the function values on these parts, its range of values is divided into intervals, and then summed with the corresponding weights of the inverse measures of these intervals.

Definition

The Lebesgue integral is determined step by step, moving from simpler to complex functions. We assume that the space given to measure $(X,{\mathcal {F}},\mu )$ ${\ displaystyle (X, {\ mathcal {F}}, \ mu)}$ $(X,\mathcal{F},\mu)$ and it defines the Borel function $f\colon (X,{\mathcal {F}})\to (\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ))$ ${\ displaystyle f \ colon (X, {\ mathcal {F}}) \ to (\ mathbb {R}, {\ mathcal {B}} (\ mathbb {R}))}$ $f\colon (X,{\mathcal {F}})\to (\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ))$ .

Definition 1. Let $f$ ${\ displaystyle f}$ $f$ - an indicator of some measurable set, that is $f(x)=\mathbf {1} _{A}(x)$ ${\ displaystyle f (x) = \ mathbf {1} _ {A} (x)}$ $f(x)={\mathbf {1}}_{A}(x)$ where $A\in {\mathcal {F}}$ ${\ displaystyle A \ in {\ mathcal {F}}}$ $A\in {\mathcal {F}}$ . Then the Lebesgue integral of the function $f$ ${\ displaystyle f}$ $f$ by definition:

\int \limits _{X}f(x)\,\mu (dx)\equiv \int \limits _{X}f\,d\mu =\mu (A).

{\ displaystyle \ int \ limits _ {X} f (x) \, \ mu (dx) \ equiv \ int \ limits _ {X} f \, d \ mu = \ mu (A).}

\int \limits _{X}f(x)\,\mu (dx)\equiv \int \limits _{X}f\,d\mu =\mu (A).

Definition 2. Let $f$ ${\ displaystyle f}$ $f$ - simple function , that is $f(x)=\sum \limits _{i=1}^{n}f_{i}\,\mathbf {1} _{F_{i}}(x)$ ${\ displaystyle f (x) = \ sum \ limits _ {i = 1} ^ {n} f_ {i} \, \ mathbf {1} _ {F_ {i}} (x)}$ $f(x)=\sum \limits _{{i=1}}^{n}f_{i}\,{\mathbf {1}}_{{F_{i}}}(x)$ where $\{f_{i}\}_{i=1}^{n}\subset \mathbb {R}$ ${\ displaystyle \ {f_ {i} \} _ {i = 1} ^ {n} \ subset \ mathbb {R}}$ $\{f_{i}\}_{{i=1}}^{n}\subset {\mathbb {R}}$ , but $\{F_{i}\}_{i=1}^{n}\subset {\mathcal {F}}$ ${\ displaystyle \ {F_ {i} \} _ {i = 1} ^ {n} \ subset {\ mathcal {F}}}$ $\{F_{i}\}_{{i=1}}^{n}\subset {\mathcal {F}}$ - final partition $X$ ${\ displaystyle X}$ $X$ on measurable sets. Then

\int \limits _{X}f(x)\,\mu (dx)=\sum \limits _{i=1}^{n}f_{i}\,\mu (F_{i})

{\ displaystyle \ int \ limits _ {X} f (x) \, \ mu (dx) = \ sum \ limits _ {i = 1} ^ {n} f_ {i} \, \ mu (F_ {i} )}

\int \limits _{X}f(x)\,\mu (dx)=\sum \limits _{{i=1}}^{n}f_{i}\,\mu (F_{i})

.

Definition 3. Now let $f$ ${\ displaystyle f}$ $f$ - non-negative function, i.e. $f(x)\geqslant 0\;\forall x\in X$ ${\ displaystyle f (x) \ geqslant 0 \; \ forall x \ in X}$ $f(x)\geqslant 0\;\forall x\in X$ . Consider all simple functions. $\{f_{s}\}$ ${\ displaystyle \ {f_ {s} \}}$ $\{f_{s}\}$ such that $f_{s}(x)\leqslant f(x)\;\forall x\in X$ ${\ displaystyle f_ {s} (x) \ leqslant f (x) \; \ forall x \ in X}$ $f_{s}(x)\leqslant f(x)\;\forall x\in X$ . We denote this family ${\mathcal {P}}_{f}$ ${\ displaystyle {\ mathcal {P}} _ {f}}$ ${\mathcal {P}}_{f}$ . For each function from this family, the Lebesgue integral is already defined. Then the integral of $f$ ${\ displaystyle f}$ $f$ is given by the formula:

\int \limits _{X}f(x)\,\mu (dx)=\sup \left\{\int \limits _{X}f_{s}(x)\,\mu (dx)\;\vert \;f_{s}\in {\mathcal {P}}_{f}\right\}

{\ displaystyle \ int \ limits _ {X} f (x) \, \ mu (dx) = \ sup \ left \ {\ int \ limits _ {X} f_ {s} (x) \, \ mu (dx ) \; \ vert \; f_ {s} \ in {\ mathcal {P}} _ {f} \ right \}}

Finally, if the function $f$ ${\ displaystyle f}$ arbitrary sign, then it can be represented as the difference of two non-negative functions. Indeed, it is easy to see that:

f(x)=f^{+}(x)-f^{-}(x),

{\ displaystyle f (x) = f ^ {+} (x) -f ^ {-} (x),}

Where

f^{+}(x)=\max(f(x),0),\;f^{-}(x)=-\min(0,f(x))

{\ displaystyle f ^ {+} (x) = \ max (f (x), 0), \; f ^ {-} (x) = - \ min (0, f (x))}

.

Definition 4. Let $f$ ${\ displaystyle f}$ - arbitrary measurable function. Then its integral is given by the formula:

\int \limits _{X}f(x)\,\mu (dx)=\int \limits _{X}f^{+}(x)\,\mu (dx)-\int \limits _{X}f^{-}(x)\,\mu (dx)

{\ displaystyle \ int \ limits _ {X} f (x) \, \ mu (dx) = \ int \ limits _ {X} f ^ {+} (x) \, \ mu (dx) - \ int \ limits _ {X} f ^ {-} (x) \, \ mu (dx)}

.

Definition 5. Finally let $A\in {\mathcal {F}}$ ${\ displaystyle A \ in {\ mathcal {F}}}$ arbitrary measurable set. Then by definition

\int \limits _{A}f(x)\,\mu (dx)=\int \limits _{X}f(x)\,\mathbf {1} _{A}(x)\,\mu (dx)

{\ displaystyle \ int \ limits _ {A} f (x) \, \ mu (dx) = \ int \ limits _ {X} f (x) \, \ mathbf {1} _ {A} (x) \ , \ mu (dx)}

,

Where $\mathbf {1} _{A}(x)$ ${\ displaystyle \ mathbf {1} _ {A} (x)}$ - indicator-function of the set $A$ ${\ displaystyle A}$ .

Example

Consider the Dirichlet function $f(x)\equiv \chi _{\mathbb {Q} _{[0,1]}}(x)$ ${\ displaystyle f (x) \ equiv \ chi _ {\ mathbb {Q} _ {[0,1]}} (x)}$ given on $([0,1],{\mathcal {B}}([0,1]),m)$ ${\ displaystyle ([0,1], {\ mathcal {B}} ([0,1]), m)}$ where ${\mathcal {B}}([0,1])$ ${\ displaystyle {\ mathcal {B}} ([0,1])}$ - Borel σ-algebra on $[0,1]$ ${\ displaystyle [0,1]}$ , but $m$ ${\ displaystyle m}$ - Lebesgue measure . This function takes on the value $one$ ${\ displaystyle 1}$ at rational points and $0$ ${\ displaystyle 0}$ in the irrational . Easy to see that $f$ ${\ displaystyle f}$ not integrable in the sense of Riemann. However, it is a simple function on a space with finite measure, because it takes only two values, and therefore its Lebesgue integral is defined and equals:

\int \limits _{[0,1]}f(x)\,m(dx)=1\cdot m(\mathbb {Q} _{[0,1]})+0\cdot m([0,1]\setminus \mathbb {Q} _{[0,1]})=1\cdot 0+0\cdot 1=0.

{\ displaystyle \ int \ limits _ {[0,1]} f (x) \, m (dx) = 1 \ cdot m (\ mathbb {Q} _ {[0,1]}) + 0 \ cdot m ([0,1] \ setminus \ mathbb {Q} _ {[0,1]}) = 1 \ cdot 0 + 0 \ cdot 1 = 0.}

Indeed, the measure of the segment $[0,1]$ ${\ displaystyle [0,1]}$ is equal to 1, and since the set of rational numbers is countable , its measure is equal to 0, which means that the measure of irrational numbers is equal to $1-0=1$ ${\ displaystyle 1-0 = 1}$ .

Remarks

Because $|f(x)|=f^{+}(x)+f^{-}(x)$ ${\ displaystyle | f (x) | = f ^ {+} (x) + f ^ {-} (x)}$ measurable function $f(x)$ ${\ displaystyle f (x)}$ Lebesgue integrable if and only if the function $|f(x)|$ ${\ displaystyle | f (x) |}$ Lebesgue integrable. This property does not hold with respect to the Riemann integral;
Depending on the choice of space, measure and function, the integral can be finite or infinite. If the integral of a function is finite, then the function is called Lebesgue integrable or summable ;
If the function is defined on a probability space $(\Omega ,{\mathcal {F}},\mathbb {P} )$ ${\ displaystyle (\ Omega, {\ mathcal {F}}, \ mathbb {P})}$ and measurable, it is called a random variable , and its integral is called the expectation or mean. A random variable is integrable if it has a finite mathematical expectation.

Properties

Lebesgue integral is linear, that is
$\int \limits _{X}[af(x)+bg(x)]\,\mu (dx)=a\int \limits _{X}f(x)\,\mu (dx)+b\int \limits _{X}g(x)\,\mu (dx)$ ${\ displaystyle \ int \ limits _ {X} [af (x) + bg (x)] \, \ mu (dx) = a \ int \ limits _ {X} f (x) \, \ mu (dx) + b \ int \ limits _ {X} g (x) \, \ mu (dx)}$ ,

Where

a,b\in \mathbb {R}

{\ displaystyle a, b \ in \ mathbb {R}}

- arbitrary constants;

Lebesgue integral preserves inequalities, that is, if $0\leqslant f(x)\leqslant g(x)$ ${\ displaystyle 0 \ leqslant f (x) \ leqslant g (x)}$ almost everywhere $f(x)$ ${\ displaystyle f (x)}$ measurable and $g(x)$ ${\ displaystyle g (x)}$ integrable then integrable and $f(x)$ ${\ displaystyle f (x)}$ and moreover
$0\leqslant \int \limits _{X}f(x)\,\mu (dx)\leqslant \int \limits _{X}g(x)\,\mu (dx)$ ${\ displaystyle 0 \ leqslant \ int \ limits _ {X} f (x) \, \ mu (dx) \ leqslant \ int \ limits _ {X} g (x) \, \ mu (dx)}$ ;
The Lebesgue integral does not depend on the behavior of the function on the set of measure zero, that is, if $f(x)=g(x)$ ${\ displaystyle f (x) = g (x)}$ almost everywhere then
$\int \limits _{X}f(x)\,\mu (dx)=\int \limits _{X}g(x)\,\mu (dx)$ ${\ displaystyle \ int \ limits _ {X} f (x) \, \ mu (dx) = \ int \ limits _ {X} g (x) \, \ mu (dx)}$ .

Lebesgue integral sums

Lebesgue integral sums for a function $f(x)$ ${\ displaystyle f (x)}$ and measures $\mu$ ${\ displaystyle \ mu}$ are called the sums of the form

S=\sum _{k=1}^{N}y_{k}\cdot \mu \{x\in X:y_{k}<f(x)\leqslant y_{k+1}\}

{\ displaystyle S = \ sum _ {k = 1} ^ {N} y_ {k} \ cdot \ mu \ {x \ in X: y_ {k} <f (x) \ leqslant y_ {k + 1} \ }}

,

Where $y_{1}<y_{2}<\dots <y_{N}$ ${\ displaystyle y_ {1} <y_ {2} <\ dots <y_ {N}}$ - splitting the range of the function $f(x)$ ${\ displaystyle f (x)}$ .

Each such sum is a Lebesgue integral of a simple function that approximates the function $f(x)$ ${\ displaystyle f (x)}$ - at each point it takes one of the values $y_{1},y_{2},\dots ,y_{N}$ ${\ displaystyle y_ {1}, y_ {2}, \ dots, y_ {N}}$ (namely, $y_{k}$ ${\ displaystyle y_ {k}}$ on a subset $\{x\in X:y_{k}<f(x)\leqslant y_{k+1}\}$ ${\ displaystyle \ {x \ in X: y_ {k} <f (x) \ leqslant y_ {k + 1} \}}$ ). Therefore, if the function $f(x)$ ${\ displaystyle f (x)}$ Lebesgue integrable, these sums converge to its integral when $y_{1}\rightarrow -\infty$ ${\ displaystyle y_ {1} \ rightarrow - \ infty}$ , $y_{N}\rightarrow +\infty$ ${\ displaystyle y_ {N} \ rightarrow + \ infty}$ and split diameter $\delta =\max\{y_{2}-y_{1},\dots ,y_{N}-y_{N-1}\}$ ${\ displaystyle \ delta = \ max \ {y_ {2} -y_ {1}, \ dots, y_ {N} -y_ {N-1} \}}$ tends to zero.

A special feature of the Lebesgue integral sums is that for their calculation it is not necessary to calculate the values of the integrable function - in fact only the distribution function of its values is needed:

F(y)=\mu \{x\in X:f(x)\leqslant y\}

{\ displaystyle F (y) = \ mu \ {x \ in X: f (x) \ leqslant y \}}

Then the Lebesgue integral sums for the function $f(x)$ ${\ displaystyle f (x)}$ and measures $\mu$ ${\ displaystyle \ mu}$ become integral Riemann-Stieltjes sums for the function $y$ ${\ displaystyle y}$ and distribution functions $F(y)$ ${\ displaystyle F (y)}$ :

S=\sum _{k=1}^{N}y_{k}(F(y_{k+1})-F(y_{k}))\rightarrow \int \limits _{-\infty }^{+\infty }ydF(y)

{\ displaystyle S = \ sum _ {k = 1} ^ {N} y_ {k} (F (y_ {k + 1}) - F (y_ {k})) \ rightarrow \ int \ limits _ {- \ infty} ^ {+ \ infty} ydF (y)}

.

If the distribution function $F(y)$ ${\ displaystyle F (y)}$ has a density of: $dF(y)=\rho (y)dy$ ${\ displaystyle dF (y) = \ rho (y) dy}$ , then Lebesgue integral sums are transformed into Riemann integral sums :

S=\sum _{k=1}^{N}y_{k}\rho (y_{k})(y_{k+1}-y_{k})\rightarrow \int \limits _{-\infty }^{+\infty }y\rho (y)dy

{\ displaystyle S = \ sum _ {k = 1} ^ {N} y_ {k} \ rho (y_ {k}) (y_ {k + 1} -y_ {k}) \ rightarrow \ int \ limits _ { - \ infty} ^ {+ \ infty} y \ rho (y) dy} .

Since distribution functions naturally arise in probability theory, statistical and quantum physics, Lebesgue integral sums are actually used to calculate the Lebesgue integral, mainly in the applications of these theories. Most often, the Lebesgue integral is calculated as the Riemann integral equal to it (in those cases when the latter makes sense).

The convergence of Lebesgue integrals of sequences of functions

Levi's monotone convergence theorem
Lebesgue's theorem on majorized convergence
Lemma veils

Notes

↑ Lebesgue, Henri (1904). "Leçons sur l'intégration et la recherche des fonctions primitives". Paris: Gauthier-Villars.

Literature

Kolmogorov A.N. , Fomin S.V. Elements of the theory of functions and functional analysis. - ed. fourth, reworked. - M .: Science , 1976 . - 544 s.
Trenogin V. А. Functional analysis. - M .: Science , 1980 . - 495 s.
Shilov G.E. Mathematical analysis. Special course. - 2nd. - M .: Fizmatlit , 1961 . - 436 s.

Frolov N. А. The theory of functions of a real variable. - 2nd. - M .: GUPIMPR , 1961 . - 173 s.

[1] Lebesgue, Henri (1904). "Leçons sur l'intégration et la recherche des fonctions primitives". Paris: Gauthier-Villars.