PropositionStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-27

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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

1.1

If $E \in A$ , then the preimage of any Borel set $B \subseteq [g i v e n, L 1] R$ 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].

givenL1

1.2

If $1_{E}$ is measurable, then

givenL1

$E = 1_{E}^{- 1} ((1 / 2, \infty)),$

and $(1 / 2, \infty)$ is a Borel subset of $R$ . Hence [L1] gives $E \in A$ . [given, L1]

2.1

Steps 1.1 and 1.2 prove the equivalence.

step 1.1step 1.2∎

Depends on

A measurable function between measurable spaces

Used by

An uncountable supremum of measurable indicators can be nonmeasurable 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
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

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)