In his notes here on modes of convergence for measurable functions $f_n : X \to \mathbb{C}$, Terry Tao provides the following definition of uniform integrability:
Let $(X,\mathcal{B},\mu)$ be a measure space (of possibly infinite measure), and let $f_n : X \to \mathbb{C}$ be a sequence of absolutely integrable functions. The $f_n$ are said to be uniformly integrable if the following three statements hold:
- (Uniform bound on $L^1$ norm) One has $$\sup_n\|f_n\|_{L^1} = \sup_n\int_X|f_n|\,d\mu < +\infty.$$
- (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 to width or horizontal infinity) For every $\epsilon > 0$, there is a finite measure subset $E$ of $X$ such that $$\int_{X \setminus E} |f_n|\,d\mu \leq \epsilon$$ for all $n$.
Are all three of these statements necessary? I believe we can prove the first from the second and third as follows: Pick $M > 0$ and $E \in \mathcal{B}$ of finite measure such that $\int_{|f_n| \geq M} |f_n|\,d\mu \leq 1$ and $\int_{X \setminus E} |f_n|\,d\mu \leq 1$ for all $n$. Then for all $n$ we can write $$\int_X |f_n|\,d\mu \leq \int_{X \setminus E} |f_n|\,d\mu + \int_{|f_n| \geq M} |f_n|\,d\mu + \int_{E \cap \{|f_n| < M\}} |f_n|\,d\mu \leq 1 + 1 + M\mu(E),$$thus obtaining a uniform bound for the $\|f_n\|_{L^1}$. Am I making a mistake in this argument, or is this definition for uniform integrability in fact redundant?