I have a sequence of functions converging pointwise a.e. on a finite measure space, [tex] \int_X |f_n|^p \leq M (1 < p \leq \infty [/tex] for all n. I need to conclude that [tex] f \in L^p [/tex] and [tex] f_n \rightarrow f [/tex] in [tex] L^t [/tex] for all [tex] 1 \leq t < p. [/tex]
The Attempt at a Solution
By Fatous I can show [tex] f \in L^p [/tex] and since [tex] L^t \subseteq L^p [/tex] for finite measure spaces, I have everything in L^t as well. I can apply Egoroffs to get [tex] \int_E |f_n-f|^t < \epsilon [/tex] with [tex] \mu(X-E) < \delta [/tex]. Any ideas on how to proceed? And thanks for your time!