Converting between bound on probability measures and densities

Question

Suppose that $P$ and $Q$ are two probability measures on the same probability space with $P(A) \leq c Q(A)$ for each (measurable) set $A$.

Is it true that $dP/dQ$ is then bounded by $c$ $P$-almost surely?

Answer 1

Let $f$ denote the Random-Nikodym derivative $\frac{dP}{dQ}$. Then $\int_Af\,dQ\leq cQ(A)$ for all $A$ and so

$$\int_A (c-f)dQ\geq0\quad\text{for all} \quad A$$

From this, it should follow that $f\leq c$ $Q$-a.s. Take for instance $A=\{f>c\}$

(You ma use a well know fact that says that if $\phi\geq0$ and $\int \phi\,d\mu=0$, then $\phi=0$ $\mu$-a.s.)