Consider a probability space $(\Omega, \mathcal{F}, P)$ with an increasing family $\{\mathcal{F}_n\}$ of sub-$\sigma$-fields of $\mathcal{F}$. Suppose there is another probability measure Q on $(\Omega, \mathcal{F})$ such that if $P_n, Q_n$ are the restrictions of $P, Q$ to $(\Omega, \mathcal{F})$, then $Q_n(A) =\int_A \Lambda_ndP_n,\; n \geq 0$ for some $\Lambda_n \geq 0$ a.s. with $E[\Lambda_n] = 1$.
Then $A(\Lambda_n, \mathcal{F}_n), n ≥ 0$, is a non-negative mean $1$ martingale under $P$. Let $E_Q[\cdot], E_P [\cdot]$ denote the expectations corresponding to $P, Q$ respectively.
Suppose $$(\star) \; \sup_n \; E_Q[\log \Lambda_n] < \infty$$ Show that under $P, \Lambda_n$ converges a.s. to a limiting random variable $\Lambda_{\infty}$ such that $E[\Lambda_{\infty}] = 1$.
I have been unable to make any progress. We have also been given a hint:
Hint: Change measure in ($\star$)
If $(X_i)$ is a family of random variables such that $E\phi (X_i)$ is bounded for some non-decreasing function $\phi$ with $\lim_{t \to \infty} \frac {\phi(t)} t=\infty$ then the family is uniformly integrable. You can find a proof of this in P A Meyer's book on Probability and Potential. The given hypothesis tarnslates to $sup_n E_P[\phi (\Lambda_n) <\infty$ where $\phi (t) =t \log t$. Now you can finish the proof using that fact that any uniformly integrable martingale converge a.s as well as in $L^{1}$.