PropositionStatement: Literature-sourcedProof: AI-adaptedverified 2026-09-23 (gpt-6-sol)

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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An indicator function is measurable exactly when its set is measurable

Statement

Let $(X, A)$ be a measurable space and let $E \subseteq X$ . The indicator function

$1_{E} (x) : = {\begin{cases} 1, & x \in E, \\ 0, & x \notin E \end{cases}$

is measurable as a map $X \to R$ if and only if $E \in A$ .

Facts & Assumptions

Given: A measurable space $(X, A)$ , a subset $E \subseteq X$ , and the indicator function $1_{E}$ .

[L1]

A function is measurable exactly when the preimage of every measurable set in the codomain is measurable in the domain. (A measurable function between measurable spaces)

Proof

technique · direct

1.1givenL1

If $E \in A$ , then the preimage of any Borel set $B \subseteq R$ under $1_{E}$ is one of $\emptyset$ , $X$ , $E$ , or $X ∖ E$ , because $1_{E}$ takes only the values $0$ and $1$ . Each of those sets lies in $A$ , so $1_{E}$ is measurable by [L1].

1.2givenL1

If $1_{E}$ is measurable, then $E = 1_{E}^{- 1} ((1 / 2, \infty))$ . The interval $(1 / 2, \infty)$ is Borel, so [L1] gives $E \in A$ .

2.1step 1.1step 1.2∎

Steps 1.1 and 1.2 prove the equivalence.

Depends on

A measurable function between measurable spaces

Used by

A hypersurface jump is not W^1,p Counterexample
A mild heat solution need not be classical at the initial time Counterexample
A step has no locally integrable weak derivative Counterexample
An uncountable supremum of measurable indicators can be nonmeasurable Counterexample
Lp and Sobolev classes do not determine point values Counterexample
Pointwise modification can destroy path continuity Counterexample
The heat equation has no finite propagation speed Counterexample
The heat flow need not converge in supremum norm Counterexample
A Lebesgue measurable function that is not Borel measurable Example
Indicator functions of measurable sets are measurable Example
The Dirichlet function is Borel measurable and nowhere continuous Example
The heat flow of an interval indicator is a difference of Gaussian tails Example
The heat smoothing time exponent is forced by scaling Example
FALSE: composing a Lebesgue measurable function with a continuous map preserves measurability False statement
FALSE: the supremum of an arbitrary family of measurable functions is always measurable False statement
Cauchy sequences in measure converge in measure Theorem
Pairwise-independent Borel-Cantelli frequency law Theorem

Dependency tree · two levels

2 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Sheldon Axler, Measure, Integration and Real Analysis, Section 2B (standard reference, not scraped)