Hi,

Reading through a proof in a book that the intersection of Lp and Lq spaces is complete, I'm wondering the following (which I think was used by the author):

In a measure space $\displaystyle (X, \mathcal{M}, \mu)$, with $\displaystyle 1 \leq p, q \leq \infty$, is it true that if $\displaystyle (f_n)$ is a sequence in $\displaystyle L^p(\mu)\cap L^q(\mu)$ such that $\displaystyle f_n\to f$ in $\displaystyle L^p(\mu)$ and $\displaystyle f_n\to g$ in $\displaystyle L^q(\mu)$, then $\displaystyle f = g$ $\displaystyle \mu$-a.e.

Since there is no relationship between the two norms for arbitrary spaces X, I do not know how to go about proving it if it were indeed correct. Your insights would be most appreciated!

Thanks