Show that $\left\| f \right\|_r$ is continuous function in $r$.

Question

I am trying to prove a statement below.

If $f\in L^1(0,\infty)\cap L^2(0,\infty)$ then

$\left\| f \right\|_r$ is continuous function in $r$ on $[1,2]$.

I just proved that $f\in L^r(0,\infty)$ $\forall r\in (1,2)$ but don't know how to prove the continuity...

Answer 1

Let $\{f_n\}$ be a sequence of simple functions such that $f_n \to f$ almost everywhere and $|f_n| \leq |f|$ almost everywhere for each $n$. Then $f_n \to f$ in $L^{1}$ as well as $L^{2}$. Claim: $\|f_n - f\|_r\to 0$ uniformly for $1 \leq r \leq 2$. Let $g_n=f_n-f$. Then $$\int |g_n|^{r} =\int_{|g_n| \leq 1} |g_n|^{r} +\int_{|g_n| > 1} |g_n|^{r} \leq \int |g_n|+\int |g_n|^{2} \to 0.$$ This proves the claim. It is trivial to check that $\|h\|_r$ is a continuous function of $r$ when $h$ is simple. By uniform convergence the same is true for the given $f$.