Let $r≥1$ be fixed and let $X_n →X$ in $L^r$ and $Y_n →Y$ in $L^r$ as $n→∞$. Show that $X_n + Y_n → X + Y$ in $L^r$ as $n → ∞$

Question

Let $r≥1$ be fixed and let $X_n →X$ in $L^r$ and $Y_n →Y$ in $L^r$ as $n→∞$.

Show that $X_n + Y_n → X + Y$ in $L^r$ as $n → ∞$.

I know that I need to show that $\mathbb{E}(|X_n+Y_n-X-Y|^r)→0$ as $n→∞$ and that $|X+Y|^r$ is integrable but I'm not sure how I'd go about that.

From the question I know that $\mathbb{E}(|X_n-X|^r)→0$ as $n→∞$ and $\mathbb{E}(|Y_n-Y|^r)→0$ but I don't know how to deal with the cross terms and I'm getting a similar problem for the integrability condition.

Answer 1

Use the inequality $$\left(a+b\right)^r\leqslant 2^{r-1}\left(a^r+b^r\right)$$ valid for nonnegative $a$ and $b$, which can be showed by convexity of $t\mapsto t^r$.

Use it with $a=\left\lvert X\left(\omega\right)\right\rvert$ and $b=\left\lvert Y\left(\omega\right)\right\rvert$ for a fixed $\omega$, then integrate to get that $X+Y$ belongs to $\mathbb L^r$.

I leave to you the choice of $a$ and $b$ for establishing the convergence of $X_n+Y_n$ to $X+Y$.