How Do Functions Converge in L^t Spaces on Finite Measure Domains?

[blinktx411]

Homework Statement

I have a sequence of functions converging pointwise a.e. on a finite measure space, \int_X |f_n|^p \leq M (1 < p \leq \infty for all n. I need to conclude that f \in L^p and f_n \rightarrow f in L^t for all 1 \leq t < p.

Homework Equations

The Attempt at a Solution

By Fatous I can show f \in L^p and since L^t \subseteq L^p for finite measure spaces, I have everything in L^t as well. I can apply Egoroffs to get \int_E |f_n-f|^t < \epsilon with \mu(X-E) < \delta. Any ideas on how to proceed? And thanks for your time!

 

[micromass]

Maybe try something like

\int_X{|f_n-f|^t}=\int_{X\setminus E}{|f_n-f|^t}+\int_{E}{|f_n-f|^t}

Try to find an upper bound K of |f_n-f|^t. Then the first integral becomes

\int_{X\setminus E}{|f_n-f|^t}\leq \mu(X\setminus E) K\leq \varepsilon K.

 

[blinktx411]

Thanks for the reply!
This is what I've been trying, but I cannot see a reason why |f_n-f|^t should be bounded on this set.

 

[micromass]

OK, here's how to proceed:

First, prove that

\lim_{\lambda \rightarrow 0}{\ \ \sup_n{\int_{\{|f_n|^t\geq \lambda\}}{|f_n|^t}}}=0

Hint: if 0<a<b, then b^t=b^{t-p}b^p\leq a^{t-p}b^p

Second, prove that for each \varepsilon >0, there exists a \delta>0 such that for each E with \mu(E)<\delta, we have

\int_E{|f_n|^t}<\varepsilon

for each n.