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An analytic subset of a Polish space is a continuous image of a Borel set, equivalently a projection of a closed subset of its product with the Baire space of sequences.

A function into a topological space is Borel measurable when the inverse image of every Borel set is measurable. Equivalently, it is measurable when the target carries its Borel sigma-algebra.

A Borel measure on a topological space is a measure defined on the sigma-algebra generated by the open sets, equivalently on the Borel sets.

A Borel set of Lebesgue measure zero. Every Lebesgue-null set is contained in a Borel null set, by outer regularity; consequently testing all Borel null sets suffices in the definition of a Sierpiński set.

The Borel sigma-algebra of a topological space $X$ is the sigma-algebra generated by its open sets. Its members are exactly the Borel sets.

The continuity set of a function is the set of points where it is continuous. For a real function $f$, define its oscillation at $x$ byThe sets $\{x:{\omega }_f(x)<\epsilon \}$ are open, and $f$ is continuous at $x$ exactly when ${\omega }_f(x)=0$. Hence its continuity set is the $G_{\delta }$ Borel set

A Euclidean body is a nonempty bounded measurable subset of ${\mathbb{R}}^n$ of positive Lebesgue measure. Projection inequalities are unchanged by modifying a body on a set of measure zero; geometric equality statements commonly impose additional regularity.

Suppose $P({\xi }^{\prime },z)$ has degree $m$ in $z$ and a nonzero constant leading coefficient $a$. Among the $m+1$ heights $h_{\ell }=3\ell$, at least one satisfiesat each real ${\xi }^{\prime }$. Factor into $m$ linear factors with multiplicity. Each root's imaginary part can be within distance less than one of at most one candidate height. The pigeonhole principle leaves a height whose distance from every root is at least one, proving the product bound. The height can be chosen measurably: the set where a candidate succeeds is the closed intersection of $|P({\xi }^{\prime },q+i{h}_{\ell })|\ge |a|$ over rational $q$, and selecting the first successful candidate gives a Borel set partition. This avoids any need for continuous global root labels.

A Lebesgue measurable set is a member of the completion of the Borel sigma-algebra with respect to Lebesgue measure. In particular, it differs from a Borel set by a null set.

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.