Statement of Vitali convergence theorem

Question

Let $f,f_n \in L^p(X,\mu)$, where $p\in[0,\infty)$ and we assume that $\mu(X)<\infty$. From the wikipedia page, the statement of Vitali convergence theorem is written as

$f_n\to f$ in $L^p$ if and only if we have

(i) $f_n$ converges in measure to $f$.

(ii) For every $\varepsilon>0$, there exists a $\delta>0$ such that if a measurable set $E$ satisfies $\mu(E)<\delta$, then for every $n$ we have
$$
\int_E |f_n|^p d\mu < \varepsilon^p.
$$

Remark: (i) and (ii) implies uniform integrability of $(|f_n|^p)_n$ whereas the uniform integrability of $(|f_n|^p)_n$ implies (ii).

This baffles me because isn't uniform integrability of $(|f_n|^p)_n$ simply equivalent to (ii)? Why do we need (i) at all?

Answer 1

Uniform integrabilty is not equivalent to ii). Definition of uniform integrabilty is $\int_{|f_n|>M} |f_n| \to 0$ uniformly in $n$ as $M \to \infty$. This is equivalent to ii) plus the condition $\sup_n \int |f_n| <\infty$