Let $X$ be a non-negative random variable in $L^p(\Omega, \mathcal{A},P)$.
We have shown that $$||\min(X,M)||_p \leq c$$ for all $M \geq 0$
and want to conclude that $$||X||_p \leq c.$$
In the lecture notes it is written that this follows by Monotonce Convergence theorem. But how does this work precisely?
So we have $$\lim_{M \rightarrow \infty} \min(X,M)=X$$
and therefore
$$||X||_p = ||\lim_{M \rightarrow \infty} \min(X,M)||_p=E[(\lim_{M \rightarrow \infty} \min(X,M))^p]^{1/p}=\lim_{M \rightarrow \infty}E[\min(X,M)^p]^{1/p} =\lim_{M \rightarrow \infty} ||\min(X,M)||_p \leq \lim_{M \rightarrow \infty} c = c $$
Is this correct? Is it clear that I can pull out the limit even though it is inside $(\cdot)^p$?
Thanks a lot!