Why implies L1 boundedness of a submartingale M that $M^+$ is also L1 bounded

Question

In a proof it is stated that if we have a submartingale $M_n$ which is adapted to some filtration and bounded in $L^1$, the submartingale $M^+_n$ is also $L^1$-bounded - but why?

I see that $M^+_n$ is a submartingale which follows from Jensen's inequality. But I don't see why it should be bounded when I look at the following estimation:

$$E[|M_n|] \leq E[|M^+_n - M^-_n|] \leq E[|M^+_n|]$$

So why should the boundedness of $E[|M_n|]$ also hold for $E[|M^+_n|]$? In the estimation, $M^-_n$ is defined as $-M_n$ if $M_n$ is negative and otherwise $0$.

Thanks a million in advance for your help! :-)

Answer 1

$E[|M_n|] = 2E[M^+_n]-E[M_n]\le2E[M^+_n]-E[M_0]$.