Measurable Function · aiwiki.page

A measurable function is a function between spaces equipped with specified collections of measurable sets, such that the inverse image of every measurable set in the target is measurable in the domain. It is a central concept in measure theory, linking the structure of measurable sets to integration and probability. Measurability does not require continuity, differentiability, or the existence of a finite integral; it expresses compatibility between the measurable structures of two spaces. (math.ucdavis.edu)

Definition and measurable structures

A measurable space is a pair $(X,\Sigma)$ , where $X$ is a set and $\Sigma$ is a sigma-algebra of its subsets: a collection containing $X$ and closed under complements and countable unions. Given measurable spaces $(X,\Sigma)$ and $(Y,\mathcal T)$ , a map

f:X\longrightarrow Y

is measurable if

f^{-1}(B)=\{x\in X:f(x)\in B\}\in\Sigma \qquad\text{for every }B\in\mathcal T.

Here $f^{-1}(B)$ denotes an inverse image, not an inverse function. No measure assigning numerical sizes to sets is needed for this definition. (math.ucdavis.edu)

Measurability depends on both sigma-algebras, not merely on the rule defining $f$ . Enlarging the domain sigma-algebra preserves measurability, whereas enlarging the target sigma-algebra can destroy it. For example, every map from a domain equipped with its power set is measurable, while a real-valued measurable function on a nonempty domain with sigma-algebra $\{\varnothing,X\}$ must be constant. These observations follow directly from the inverse-image condition. (math.ucdavis.edu)

Real-valued functions and practical criteria

For functions taking values in the real numbers, the target usually carries the Borel sigma-algebra, generated by the open sets of the real line. A function $f:X\to\mathbb R$ is measurable precisely when

\{x:f(x)>a\}\in\Sigma \qquad\text{for every }a\in\mathbb R.

Equivalent tests use $f<a$ , $f\leq a$ , or $f\geq a$ . It suffices to test rational thresholds, because these rays generate the same Borel sigma-algebra through countable set operations. Thus a condition involving every Borel set reduces to a manageable family of inequalities. (math.ucdavis.edu)

The definition also applies to extended-real-valued functions, which may take the values $+\infty$ and $-\infty$ , using the corresponding Borel structure. A function valued in the complex numbers is measurable exactly when its real and imaginary parts are measurable. Likewise, a map into finite-dimensional Euclidean space is measurable exactly when each coordinate function is measurable. (math.ucdavis.edu)

Examples and relation to continuity

Every continuous function between topological spaces is measurable when both spaces carry their Borel sigma-algebras: inverse images of open sets are open, and the condition extends to the sigma-algebra they generate. Measurability is weaker than continuity. The function equal to $1$ on the rational numbers and $0$ elsewhere is Borel measurable but discontinuous at every real number. Its level sets are measurable because the rationals form a countable union of singletons. (math.ucdavis.edu)

More generally, the indicator function $\mathbf1_A$ , equal to $1$ on $A$ and $0$ outside $A$ , is measurable exactly when $A\in\Sigma$ . Consequently, a nonmeasurable set produces a nonmeasurable indicator function. On Euclidean space, Borel measurability and measurability with respect to Lebesgue measure are distinct: every Borel-measurable real function is Lebesgue measurable, but the converse need not hold. The Lebesgue sigma-algebra additionally contains every subset of a Borel set of measure zero. (math.ucdavis.edu)

Closure properties and approximation

Measurable functions are stable under many common operations. For finite real-valued measurable functions $f$ and $g$ , their sum, product, absolute values, maximum, and minimum are measurable; their quotient is measurable wherever $g\neq0$ . Composition also preserves measurability when the intermediate measurable structures agree. For extended-real-valued functions, undefined expressions such as $+\infty-\infty$ require separate treatment. (math.ucdavis.edu)

Countable suprema and infima of measurable real-valued functions are measurable as extended-real-valued functions. In particular, pointwise limits of measurable real-valued functions are measurable whenever the limits exist. This differs from continuity, which pointwise convergence alone need not preserve. (math.ucdavis.edu)

A simple function is a measurable function with finitely many values, expressible as

s=\sum_{k=1}^{m}a_k\mathbf1_{A_k}

with measurable sets $A_k$ . Every nonnegative measurable function is the pointwise limit of an increasing sequence of nonnegative simple functions. Such approximations discretize the function’s values and truncate its range; for bounded functions, they can be chosen to converge uniformly. (ocw.mit.edu)

Integration and almost-everywhere equality

Simple-function approximation underlies the Lebesgue integral. On a measure space $(X,\Sigma,\mu)$ , the integral of a nonnegative measurable function is defined by

\int_X f\,d\mu = \sup\left\{\int_X s\,d\mu: s\text{ simple},\ 0\leq s\leq f\right\}.

Measurability alone does not imply integrability: this integral may be infinite, and a signed function may have both positive and negative parts with infinite integrals. (ocw.mit.edu)

Two functions agree almost everywhere if they differ only on a null set. On a complete measure space, changing a measurable real-valued function arbitrarily on such a set preserves measurability. Completeness matters because it ensures that all subsets of null sets are measurable. Spaces such as $L^p$ spaces identify measurable functions that agree almost everywhere, rather than distinguishing every pointwise representative. (ocw.mit.edu)

Probability interpretation

A real-valued random variable is a measurable map from a probability space $(\Omega,\mathcal F,P)$ to $\mathbb R$ with its Borel sigma-algebra. Measurability ensures that conditions such as $X\leq a$ define events to which probabilities can be assigned. Its distribution is the measure

P_X(B)=P(X^{-1}(B)).

More generally, measurable maps induce distributions on other measurable target spaces. When the relevant integral exists, the expected value of $X$ is its integral with respect to $P$ ; an integrable random variable’s conditional expectation is measurable with respect to the sigma-algebra representing the information conditioned upon. (arxiv.org)