= Cumulative measure of a finite-measure set
{title2=$H_E(t)=m(E\cap(-\infty,t])$}
For a <Lebesgue measurable set> $E\subseteq\mathbb R$ of finite <measure>, the displayed <function> satisfies $|H_E(t)-H_E(s)|\leq|t-s|$. Indeed, for $s<t$, <finite additivity> and <monotonicity of a measure> give $H_E(t)-H_E(s)=m(E\cap(s,t])\leq t-s$. Thus $H_E$ is a nondecreasing <Lipschitz function> and is <uniformly continuous> even if $E$ is unbounded.
<Continuity from above of a measure> gives $H_E(t)\to0$ as $t\to-\infty$, while <continuity from below of a measure> gives $H_E(t)\to m(E)$ as $t\to\infty$. Every strictly intermediate value is attained by the <intermediate value theorem>. The corresponding half-line cut of $E$ realizes any prescribed <measure> between zero and $m(E)$; the endpoints are realized by the <empty set> and $E$ itself.
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