Assume we have a measurable space $(\Omega,\mathcal{F})$ with two probability measures $Q$ and $P$, which are equivalent, i.e. for all $A \in \mathcal{F}$ we have $$Q(A)=0 \iff P(A)=0.$$
I will denote expectations with respect to $P$ as $E_P[\cdot]$ and expectations with respect to $Q$ as $E_Q[\cdot]$.
Let $(f_n)_{n \in \mathbb{N}}$ be a sequence of non-negative random variables such that $\sup_{n \in \mathbb{N}}E_Q[f_n] < \infty$, i.e. $(f_n)_{n \in \mathbb{N}}$ is bounded in $L^1(Q)$. Assume that de Radon-Nikodyim derivative satisfies $$\frac{dQ}{dP} \in L^{\infty}(P).$$
I want to show that the sequence $(f_n)_{n \in \mathbb{N}}$ is also bounded in $L^0(P)$, i.e. that the following holds $$\sup_{n \in \mathbb{N}}P[f_n > K] \rightarrow 0, \text{ for } K \rightarrow \infty.$$
I cannot seem to get to this conclusion. Does anyone have a hint or a complete proof? Thanks a lot in advance!
Since $\mathbb{P}$ and $\mathbb{Q}$ are equivalent probability measures, there exists $g \in L^1(\mathbb{Q})$, $g \geq 0$, such that $d\mathbb{P} = g \, d\mathbb{Q}$. The integrability of $g$ implies that $g$ is uniformly integrable in the sense that for any $\epsilon>0$ there exists $\delta>0$ such that
$$A \in \mathcal{F}, \mathbb{Q}(A) \leq \delta \implies \int_A g \, d\mathbb{Q} \leq \epsilon.\tag{1}$$
Fix some $\epsilon>0$ and choose $\delta>0$ according to $(1)$. Since
$$\sup_{n \in \mathbb{N}} \mathbb{Q}(f_n>K) \leq \frac{1}{K} \sup_{n \in \mathbb{N}} \mathbb{E}_{\mathbb{Q}}(f_n)$$
we can choose $K>0$ sufficiently large such that
$$\sup_{n \in \mathbb{N}} \mathbb{Q}(f_n>K) \leq \delta$$
which implies by $(1)$ that
$$\sup_{n \in \mathbb{N}} \mathbb{P}(f_n>K) = \sup_{n \in \mathbb{N}} \int_{\{f_n>K\}} g \, d\mathbb{Q} \leq \epsilon.$$
As $\epsilon>0$ is arbitrary, this proves the assertion.