C Convex functions of Radon-Nikodym derivatives
Let \(f : \mathbb {R} \to \mathbb {R}\) be convex on \([0, +\infty )\). Remark that \(f(0)\) is finite.
We define \(f'(\infty ) := \limsup _{x \to + \infty } f(x)/x\). This can be equal to \(+\infty \) (but not \(-\infty \)).
If \(\mu \) and \(\nu \) are two finite measures and \(f'(\infty ) {\lt} \infty \), then \(x \mapsto f\left(\frac{d\mu }{d\nu }(x)\right)\) is \(\nu \)-integrable.
Proof
By convexity and \(f(0) {\lt} \infty \), \(f'(\infty ){\lt}\infty \), we can sandwich \(f\) between two affine functions of \(\frac{d\mu }{d\nu }\). Those functions are integrable since the measures are finite.
For \(f\) convex, \(f\left(\mu [g \mid m]\right) \le \mu [f \circ g \mid m]\).
Proof