Measurable set ($A\in \mathcal{F}$)

A subset of $X$ is measurable relative to a sigma-algebra $\mathcal{F}$ when it belongs to $\mathcal{F}$. A measure defined on $\mathcal{F}$ assigns sizes to these sets. Measurability depends on the chosen sigma-algebra; for example, a Lebesgue measurable set need not be a Borel set.