Let $(\Omega, F, F_t, P)$ be a filtered probability space and $(L_n)_{n \geq0}$ a family of positive and $F_t$ adapted random variables.
I have to find the conditions for which $Q_n$, defined on $F_n$, as $Q_n(A_n)=E(L_n 1_{A_n})$ is a probability measure.
Then I have to show that $Q_n(A_p)=Q_p(A_p)$ ($A_p \in F_p$, $p<n$) if and only if L_n ($n \geq 1$) is a martingale
My attempt:
Is it sufficient to impose that $E[L_n]=1$? On the second part
the if part (if L_n is a martingale) is easy to prove:
$Q_n(A_p)=E[E[L_n 1_{A_p}|F_p]]=E[L_p 1_{A_p}] = Q_p(A_p)$
how to prove is the only one condition??
It is sufficient and necessary that ${\rm E}[L_n]=1$ in order for $Q_n$ to be a probability measure on $F_n$.
Assume now that $Q_n$ is a probability measure, i.e. $L_n$ is integrable. Suppose also that $Q_n(A)=Q_p(A)$ holds for all $A\in F_p$ (for fixed $p<n$). That is, ${\rm E}[L_n\mathbf{1}_A]={\rm E}[L_p\mathbf{1}_A]$ holds for all $A\in F_p$. Since $L_p$ is integrable and measurable with respect to $F_p$ this means that ${\rm E}[L_n\mid F_p]=L_p$.