Not sure if this is the right place to post such a rookie question, but I'd appreciate some quick clarification. Is there such a term as a "Borel measurable set"? I've seen it used all over the place but I'm pretty sure it simply means a "Borel set" most of the time, which does not necessarily invoke the notion of a measure. Is this a well-defined term and if so, how is it different from a "Borel set"?
2026-03-25 17:26:12.1774459572
Is there such a term as a "Borel measurable set"?
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Yes, "Borel measurable" normally means the same thing as "Borel".
This is related to terminology from abstract measure theory, where you have a set $X$ equipped with a $\sigma$-algebra $\mathcal{F} \subset \mathcal{P}(X)$. If $A \in \mathcal{F}$, we commonly say that $A$ is "measurable" (if it's understood from context what $\sigma$-algebra we are using) or "$\mathcal{F}$-measurable" (if it's not). So "Borel measurable" is like saying "$\mathcal{F}$-measurable, where $\mathcal{F}$ is the Borel $\sigma$-field".