3
$\begingroup$

Let $X$ be a measure space and let $f_{n}$ be a sequence of functions which converge pointwise to a function $f$ in $L^{p}(X)$ where $p>1$ and suppose $g_{n}$ is a sequence of functions which converge pointwise to a function $g$ in $L^{\frac{p}{p-1}}(X)$. Prove that:

$$ \lim_{n \to \infty} \int_{X} f_{n}(x) g_{n}(x) dx = \int_{X} f(x)g(x) dx.$$

No idea how to proceed. Any help? I guess somewhere we need Hölder and DCT.

  • 0
    Can you prove it if we instead had $f_n\to f$ in $L^p$ and $g_n\to g$ in $L^{p/(p-1)}$?2010-11-04
  • 0
    Now I see that the above comment regarding norm convergence is covered by OP's other question http://math.stackexchange.com/questions/11028/convergence-of-integrals-in-lp2012-01-16

1 Answers 1