Suppose that $P$ and $Q$ are two probability measures such that $$ P(A) \leq cQ(A) + c' $$ where $$ c' = \sup_{A}\Big( P(A) - cQ(A) \Big). $$ When $c' = 0$, it is easy to see that the Radon-Nikodym derivative exists and $dP/dQ \leq c$ $P$-a.s. My question is:
When $c' > 0$, can we characterize this any further? For example, can we say that $dP/dQ \leq c$ except for a set of $P$-measure bounded by $c'$?
What I know is that if the supremum is attained then excluding $A$, we can see that on $X \setminus A$ (the complement) $dP/dQ$ is bounded by $c$ almost surely $P$. But can we say anything more?
Any signed measure $\mu$, can be decomposed uniquely as as $\mu=\mu_+-\mu_-$ where $\mu_+$ and $\mu_-$ mutually singular are positive measures.
$$\mu_+(A)=\sup\{\mu(B): B\subset A\},\quad \mu_-(A)=\sup\{-\mu(B):B\subset A\}$$
(supremum over measurable sets of course)
In your case,
$$c'=(P-c Q)_+(X)$$
In particular, when c'=0, that means that $P_-cQ\leq0$, that is $P-cQ=-(P-cQ)_-$. This in turn shows that $P\ll Q$ and $\frac{dP}{dQ}\leq c$.
If $c’>0$ there is nothing to say about the density of the absolutely continuous component of $P$ with respect to $Q$. In fact $P$ and $Q$ may even be mutually singular.
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All this comes under the name of Hahn-Jordan decomposition. The measure $|\mu|=\mu_++\mu_-$ is the variation measure of $|\mu|$.