$1\le p \lt \infty$ and $f_k$ nonnegative increasing. Then $f_k\to f$ in $L_p$ iff $\sup_k||f_k||_p \lt \infty$.

Question

Let $1\le p \lt \infty$ and $0\le f_k$ increasing to $f$, and $f_k$ measurable.

Then $f_k\to f$ in $L_p$ if and only if $\sup_k||f_k||_p \lt \infty$.

I was able to show the if part, but I can't show the only if part. How can I show this. I would greatly appreciate any help.

Answer 1

By the monotone convergence theorem, you always will have $$\int f^p = \lim_{k \rightarrow \infty} \int f_k^p$$ Taking $p$th roots (and using continuity of the function $x \rightarrow x^{1 \over p}$), one therefore has $$||f||_p = \lim_{k \rightarrow \infty} ||f_k||_p \tag 1$$ Since the $f_k$ increase to $f$, the limit in $(1)$ is an increasing limit. So one also has $$||f||_p = \sup_k ||f_k||_p \tag 2$$ If each $f_k \in L^p$ and $\lim_{k \rightarrow \infty} ||f_k - f||_p = 0$, which I'm taking as your assumption, then $f \in L^p$ since by the triangle inequality one has $||f||_p \leq ||f - f_k||_p + ||f_k||_p$. Hence $(2)$ gives that $\sup_k ||f_k||_p = ||f||_p$ is finite.

(You do need each $f_k \in L^p$ for this; otherwise you could just take $f_k = f$ to be equal to some non-$L^p$ function for all $k$ and the problem will be false).

Answer 2

Well, fk is a sequence of increasing functions to f, so with sufficient work you should be able to show that sup(k) ||fk||p = ||f||p.

Answer 3

Oh, I just read your updated version of the question.

If fk->f in Lp, we know that f is in Lp. Considering fk is a non-negative increasing sequence of functions, to converge at f it must be that sup(k) ||fk||p=||f||p. f is in Lp, so ||f||p is finite.