This is an exercise from tao's blog.
A sequence $f_n:X\to\mathbf{C}$ of absolutely integrable functions is said to be uniformly integrate if the following three statements hold:
- (Uniform bound on $L^1$ norm) One has $\sup_n\|f_n\|_{L^1(\mu)}=\sup_n\int_X |f_n|\,d\mu<\infty$.
- (No escape to vertical infinity) One has $\sup_n\int_{|f_n|\geq M}|f_n|\,d\mu\to 0$ as $M\to \infty$.
- (No escape with width infinity) One has $\sup_n\int_{|f_n|\leq\delta}|f_n|\,d\mu\to 0$ as $\delta\to 0$.
Show that every dominated sequence of measurable functions is uniformly integrable.
Let $(f_n)_{n=1}^\infty$ is a sequence of measurable function dominated by some absolutely integrable function $g$, that is $|f_n|\leq g$ for all $n=1,2,\dots$. The first statement is trivial. Note that
$$\int_{|f_n|\geq M}|f_n|\,d\mu\leq \int_{|f_n|\geq M}g\,d\mu\leq\int_{|g|\geq M}g\,d\mu,$$
then using dominated convergence theorem, the second statement follows. Now I have trouble in verifying the third statement, since $g\leq\delta$ is contained in $|f_n|\leq\delta$, the above argument doesn't work.
Best Answer
For arbitrary $m \in \Bbb{N}$, let $$ g_m := \sup_{n \in \Bbb{N}} (1_{|f_n| \leq 1/m} \cdot |f_n|). $$ Note that this is a measurable function with $0 \leq g_m \leq 1/m$, so that $g_m \to 0$ almost everywhere as $m \to \infty$.
Furthermore, it is easy to see $0 \leq g_m \leq \sup_n |f_n| \leq g$.
Thus, the dominated convergence theorem implies $$ \sup_n \int_{|f_n|\leq 1/m}|f_n| \, d\mu \leq \int \sup_n 1_{|f_n|\leq 1/m} \cdot |f_n| \, d\mu = \int g_m \, d\mu \xrightarrow[m\to\infty]{} 0. $$
By monotonicity (or since we can replace $(1_m)_m$ by any null-sequence), this implies the claim.