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Let $\mu, \nu$ be two positive Radon measures on $\mathbb{R}^n$, and $f, g: \mathbb{R}^n \to (0, \infty)$ be such that $$f\mu \le g\nu,\text{ i. e. }\int_E f(x)\mathrm{d}\mu(x) \le \int_E g(x)\mathrm{d}\nu(x)\text{ for every Borel set }E\subset\mathbb{R}^n.$$ Is it true that if $h:(0, \infty) \to (0, \infty)$ is an increasing function then $$ (h\circ f) \mu \le (h\circ g)\nu?$$

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  • $\begingroup$ Any assumptions on $f,g$ so that $h\circ f, h\circ g$ are measurable or integrable? $\endgroup$ – Dimitris Jul 30 '16 at 16:15
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No. Say $m$ is Lebesgue measure. Let $\mu=3m$, $\nu=m$, $f=1$, $g=4$. Then $f\mu<g\nu$.

Let $h$ be an increasing function with $1\le h\le 2$. Then $$ (h\circ f)\mu\ge\mu > 2\nu\ge(h\circ g)\nu.$$

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