In René Schilling's Measures, Integrals and Martingales, 2nd edition, Example 13.14 states:
Let $(X,\mathscr A,\nu)$ be a measure space and $w\in\mathcal L^1(\nu),w\geqslant0$.
(i) Taking $\mu:=w\nu\,\big/\int w\,d\nu$ in Theorem 13.13(i)...
Theorem 13.13 is Jensen's inequality, and it requires $\mu$ to be some probability measure. Thus, the $\mu$ defined above is supposed to be a probability measure constructed from $\nu$. Is it? I'm not seeing this. In particular, measures need to map sets in a sigma algebra to a number in $[0,\infty].$ The $w$ in the above definition of $\mu$ is the problem. It is a map $w:X\to[0,\infty]$.
In particular, let $A\in\mathscr A$. How is $\mu(A)$ defined? $\nu(A) \big/\int w\,d\nu$ makes sense because it is a number in $[0,\infty]$, but appending the $w$ means we need to feed $w$ a value in $X$, or else $\mu$ returns functions and not numbers.
For context, Theorem 13.13 involves the integral $\int u\,d\mu$, and Example 13.14 with claims that its proposed probability measure $\mu$ turns this integral into $\frac{\int uw\,d\nu}{\int w\,d\nu}$.
$\mu$ is supposed to be defined by $\mu (E)=\frac 1 {\int w d\nu} \int_Ew d\nu$. The correct notation for this is $d\mu=\frac {w d\nu} {\int w d\nu} $ so either there is an abuse of notation or a typo in the definition . With this definition it is clear that $\mu$ is a probability measure.