I am trying to collect a list of counterexamples about the different modes of convergence in measure theory.
By definition,
The sequence of functions ${f_n}$ converges in the measure to the function $f$ if $\lim_{n\to\infty} ||f_n (x)-f(x)||=0$.
The sequence ${f_n}$ converges almost uniformly to the function $f$ if $\forall \epsilon > 0$ there exists a measurable set E such that $\mu (E)\leq \epsilon $ and $ f_n(x)$ converges uniformly to $f$ in the complement of E.
It can be shown that the latter implies the former, but can someone provide some examples that would show that the converse is not generally try?