# 2.34

Exercise Let ${\left|f_{n}\right|\le g\in L^{1}}$ and ${f_{n}\rightarrow f}$ in measure. Show that,

• (a) ${\int f=\lim_{n\rightarrow\infty}\int f_{n}}$.
• (b) ${f_{n}\rightarrow f}$ in ${L^{1}}$.

Proof: ~

• (a) In Exercise 33 we proved a theorem analogous to Fatou’s lemma for ${f_{n}\rightarrow f}$ in measure. We will use this analogue of Fatou’s lemma to prove an analogue of dominated convergence theorem. By taking real and imaginary parts, we can assume that ${f_{n}}$ and ${f}$ are real-valued. Applying Exercise 33 to ${g+f_{n}}$ yields,

$\displaystyle \int f\le\liminf\int f_{n}$

And applying Exercise 33 to ${g-f_{n}}$ yields,

$\displaystyle \limsup\int f_{n}\le\int f$

Therefore we have ${\int f=\lim\int f_{n}}$. Note that this almost exactly the same as the normal proof of dominated convergence theorem from Fatou’s lemma.

• (b) It is easy to see that we have ${\left|f_{n}-f\right|\rightarrow0}$ in measure and ${\left|f_{n}-f\right|\le\left|f\right|+\left|f_{n}\right|\le2g}$ a.e. Therefore we apply (a) to get that ${\lim_{n\rightarrow\infty}\int\left|f_{n}-f\right|=0}$ so ${f_{n}\rightarrow f}$ in ${L^{1}}$.