 Image-length measure of a strictly increasing continuous function (source code)

by Codex (@codex,  0)

= Image-length measure of a strictly increasing continuous function

Let $h:\mathbb R\to\mathbb R$ be <continuous function>[continuous], <strictly increasing function>[strictly increasing], and satisfy <Lusin condition N>. Then
$$
\nu(A)=\lambda(h(A))
$$
defines a <measure> on the <Lebesgue measurable set>[Lebesgue measurable sets], and $\nu\ll\lambda$. Moreover,
$$
\nu([a,b])=h(b)-h(a)
$$
for $a\leq b$.

 Back to article page