Measurable function

Measurable functions represent a natural class of functions connecting spaces with distinguished algebras of sets , in particular, measurable spaces .

Content

Definition

Let be $(X,{\mathcal {F}})$ ${\ displaystyle (X, {\ mathcal {F}})}$ and $(Y,{\mathcal {G}})$ ${\ displaystyle (Y, {\ mathcal {G}})}$ - two sets with distinguished algebras of subsets . Then the function $f:X\to Y$ ${\ displaystyle f: X \ to Y}$ called ${\mathcal {F}}/{\mathcal {G}}$ ${\ displaystyle {\ mathcal {F}} / {\ mathcal {G}}}$ - measurable , or simply measurable , if the full preimage of any set of ${\mathcal {G}}$ ${\ displaystyle {\ mathcal {G}}}$ belongs ${\mathcal {F}}$ ${\ displaystyle {\ mathcal {F}}}$ , i.e

\forall B\in {\mathcal {G}},\;f^{-1}(B)\in {\mathcal {F}},

{\ displaystyle \ forall B \ in {\ mathcal {G}}, \; f ^ {- 1} (B) \ in {\ mathcal {F}},}

Where $f^{-1}(B)$ ${\ displaystyle f ^ {- 1} (B)}$ means the full prototype of the set $B$ ${\ displaystyle B}$ .

Remarks

If a $X$ ${\ displaystyle X}$ and $Y$ ${\ displaystyle Y}$ - topological spaces , and algebras ${\mathcal {F}}$ ${\ displaystyle {\ mathcal {F}}}$ and ${\mathcal {G}}$ ${\ displaystyle {\ mathcal {G}}}$ are not explicitly indicated, it is assumed that these are Borel σ-algebras of the corresponding spaces.
The meaning of this definition is that if on the set $X$ ${\ displaystyle X}$ given measure, then this function induces (passes) this measure and to the set $Y$ ${\ displaystyle Y}$ .

Real-valued measurable functions

Let function be given $f:(X,{\mathcal {F}})\to (\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ))$ ${\ displaystyle f: (X, {\ mathcal {F}}) \ to (\ mathbb {R}, {\ mathcal {B}} (\ mathbb {R}))}$ . Then the definition of measurability given above is equivalent to any of the following:

Function $f$ ${\ displaystyle f}$ measurable if
$\forall c\in \mathbb {R} ,\;\{x\in X\mid f(x)\leq c\}\in {\mathcal {F}}$ ${\ displaystyle \ forall c \ in \ mathbb {R}, \; \ {x \ in X \ mid f (x) \ leq c \} \ in {\ mathcal {F}}}$ .

Function $f$ ${\ displaystyle f}$ measurable if
$\forall a,b\in \mathbb {R}$ ${\ displaystyle \ forall a, b \ in \ mathbb {R}}$ such that $a\leq b$ ${\ displaystyle a \ leq b}$ we have $\{x\in X\mid f(x)\in |a,b|\}\in {\mathcal {F}}$ ${\ displaystyle \ {x \ in X \ mid f (x) \ in | a, b | \} \ in {\ mathcal {F}}}$ ,

Where

|a,b|

{\ displaystyle | a, b |}

denotes any interval, open, half-open or closed.

Related definitions

Let be $(X,{\mathcal {F}})=(\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ))$ ${\ displaystyle (X, {\ mathcal {F}}) = (\ mathbb {R}, {\ mathcal {B}} (\ mathbb {R}))}$ and $(Y,{\mathcal {G}})=(\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ))$ ${\ displaystyle (Y, {\ mathcal {G}}) = (\ mathbb {R}, {\ mathcal {B}} (\ mathbb {R}))}$ - two copies of the real line together with its Borel σ-algebra . Then the measurable function $f:(\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ))\to (\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ))$ ${\ displaystyle f: (\ mathbb {R}, {\ mathcal {B}} (\ mathbb {R})) \ to (\ mathbb {R}, {\ mathcal {B}} (\ mathbb {R}) )}$ called Borel .
Measurable function $f:(\Omega ,{\mathcal {F}})\to (Y,{\mathcal {G}})$ ${\ displaystyle f: (\ Omega, {\ mathcal {F}}) \ to (Y, {\ mathcal {G}})}$ where $\Omega$ ${\ displaystyle \ Omega}$ - a set of elementary outcomes , and ${\mathcal {F}}$ ${\ displaystyle {\ mathcal {F}}}$ - The σ-algebra of random events is called a random element . A particular case of a random element is a random variable for which $(Y,{\mathcal {G}})=(\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ))$ ${\ displaystyle (Y, {\ mathcal {G}}) = (\ mathbb {R}, {\ mathcal {B}} (\ mathbb {R}))}$ .

Examples

Let be $f:\mathbb {R} \to \mathbb {R}$ ${\ displaystyle f: \ mathbb {R} \ to \ mathbb {R}}$ - continuous function . Then it is measurable with respect to the Borel σ-algebra on the number line.
Let be $f:(X,{\mathcal {F}})\to (\mathbb {R} ,{\mathcal {B}}(\mathbb {R} )),$ ${\ displaystyle f: (X, {\ mathcal {F}}) \ to (\ mathbb {R}, {\ mathcal {B}} (\ mathbb {R})),}$ and $f(x)=\mathbf {1} _{A}(x),\;x\in X$ ${\ displaystyle f (x) = \ mathbf {1} _ {A} (x), \; x \ in X}$ - set indicator $A\not \in {\mathcal {F}}.$ ${\ displaystyle A \ not \ in {\ mathcal {F}}.}$ Then the function $f$ ${\ displaystyle f}$ not measurable.

Properties

Luzin's theorem . Function $f\colon \mathbb {R} \to \mathbb {R}$ ${\ displaystyle f \ colon \ mathbb {R} \ to \ mathbb {R}}$ measurable if and only if for any $\varepsilon >0$ ${\ displaystyle \ varepsilon> 0}$ there is a continuous function $h\colon \mathbb {R} \to \mathbb {R}$ ${\ displaystyle h \ colon \ mathbb {R} \ to \ mathbb {R}}$ different from $f$ ${\ displaystyle f}$ on the set of measures not more $\varepsilon$ ${\ displaystyle \ varepsilon}$ .

History

In 1901, the French mathematician A. Lebesgue , on the basis of the theory of Lebesgue integral constructed by him, set the task: to find a class of functions that is wider than analytical, but at the same time allowing the application of many analytical methods to it. By this time, there was already a general theory of measure , developed by E. Borel (1898), and Lebegue’s first works relied on Borel theory. In Lebesgue’s dissertation (1902), measure theory was generalized to the so-called Lebesgue measure . Lebesgue defined the concepts of measurable sets, bounded measurable functions and integrals for them, proved that all the “ordinary” bounded functions studied in analysis are measurable, and that the class of measurable functions is closed with respect to the basic analytic operations, including the passage to the limit . In 1904, Lebesgue summarized his theory, removing the condition of boundedness of the function.

Research Lebesgue found a wide scientific response, they were continued and developed by many mathematicians: E Borel, M. Rees , J. Vitali , MR R. Frechet , N. N. Luzin , DF Egorov, and others. The concept of convergence was introduced measure (1909), the topological properties of a class of measurable functions are deeply studied.

Lebesgue's works had another important conceptual meaning: they were completely based on the Cantor theory of sets that was controversial in those years, and the fruitfulness of Lebesgue's theory served as a weighty argument for adopting set theory as the foundation of mathematics.

Literature

A. N. Kolmogorov, S. V. Fomin. Elements of the Theory of Functions and Functional Analysis, 4th ed., M .: Nauka, 1976, 544 p.
F. Medvedev. On the history of the concept of a measurable function. // Historical and mathematical research . - M .: Fizmatgiz , 1959. - № 12 . - p . 481-492 .
Halmos P. Theory of Measure. M .: Publishing house of foreign literature, 1953.