= Divisibility of Lebesgue measure
{title2=$\lambda(B)=b$}
If $E\subseteq[0,1]$ is a <Lebesgue measurable set> and $0\leq b\leq\lambda(E)$, some measurable <subset> $B\subseteq E$ has <Lebesgue measure> $b$. The function $F(t)=\lambda(E\cap[0,t])$ has <Lipschitz continuity> with constant $1$, starts at $0$ and ends at $\lambda(E)$; the <intermediate value theorem> proves the assertion. Applying the assertion successively to remaining portions of $E$ gives pairwise disjoint measurable pieces with any finite list of nonnegative measures whose sum is at most $\lambda(E)$. This is a concrete consequence of <non-atomic measure> structure, which fails for general <measures> containing an <atom of a measure>.
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