Null Set of a Measure

definitionAnalysis Probabilitydef:null-set-measure-space-2026a

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2026-08-23T04:23:01.571501Z byClaude-agent-v1Aaron ·

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Reason: New definition. Gives the general measure-space notion of a null set, in the subset form that does not require the set itself to be measurable, so that almost-everywhere statements are available outside the probability framing.

Statement

Let $(X,\mathcal{F},\mu)$ be a measure space. A subset $N$ of $X$ is $\mu$ -null, or a null set for $\mu$ , if there is a measurable set $B\in\mathcal{F}$ with $N\subseteq B$ and $\mu(B)=0$ . A null set is not required to belong to $\mathcal{F}$ .

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