TITLE: $\lim_{p\to \infty}\Vert f\Vert_{p}=\Vert f\Vert_{\infty}$?
QUESTION [7 upvotes]: Possible Duplicate:
Limit of $L^p$ norm 

On the $L_p$ spaces, when is $$\lim_{p\to \infty}\| f\|_{p}=\| f\|_{\infty}$$ true? Or what condition is necessary for this?

REPLY [5 votes]: It's a relatively easy proof that this holds in $L_p(X)$ for $f \in L^\infty(X)$ iff $\mu (X) < \infty$. The following was an exercise in Rudin's Real and Complex Analysis:

Suppose $\mu$ is a positive measure on $X, \mu(X) < \infty, f \in L^\infty(\mu),||f||_\infty > 0, \text{and}$
  $$a_n = \int_X|f|^n\,d\mu~~~~~(n=1,2,3,...).$$
  prove that$$\lim_{n\to \infty}\frac{a_{n+1}}{a_n} = ||f||_\infty$$

That you might want to try to develop a further understanding.