Lp convergence involving powers

Question

Let $(X,\mu)$ be a finite measure space. Suppose $\int_X\lvert f_n-f\rvert^p\,d\mu\to0$ for $p\geq1$. For $q\in(0,p]$ does it hold that $$\int_X\lvert\lvert f_n\rvert^q-\lvert f\rvert^q\rvert^{\frac{p}{q}}\,d\mu\to0\quad?$$

Answer 1

Yes. I will assume that we work on a probability space. From item 6 of this answer, we know that convergence to $0$ in $\mathbb L^1$ of a sequence of random variables $\left(X_n\right)_{n\geqslant 1}$ is equivalent to its convergence in probability combined with uniform integrability. Define $$ X_n:= \lvert\lvert f_n\rvert^q-\lvert f\rvert^q\rvert^{\frac{p}{q}}. $$ By the assumption, $f_n\to f$ in probability hence by item 3 of this answer, $X_n\to 0$ in probability. For uniform integrability, we bound $X_n$ using the inequality $$ \lvert\lvert a\rvert^q-\lvert b\rvert^q\rvert^{\frac{p}{q}} \leqslant \lvert\lvert a\rvert^q+\lvert b\rvert^q\rvert^{\frac{p}{q}} \leqslant 2^{p/q-1}\left(\lvert\lvert a\rvert^p+\lvert\lvert b\rvert^p\right). $$ In this way, we are reduced to show the uniform integrability of the sequence $\left(\lvert\lvert f_n\rvert^p\right)_{n\geqslant 1}$, which is a consequence of $\int_X\lvert f_n-f\rvert^p\,d\mu\to0$.