Convergence Counterexamples
Solution 1:
a.e. $\nRightarrow $ uniformly : $f_n(x)=x^n\chi_{[0,1]}$. It converges a.e. to 0 but not uniformly.
a.e.$\nRightarrow$ $L^1$: $f_n(x)=\chi_{[n,n+1]}$ converges a.e. to 0, but the integrals are all 1.
a.e.$\nRightarrow$ in measure: use the above example: $\mu(\{f_n\geq\varepsilon\})=\mu(\{f_n=1\})=1\nrightarrow 0$
in measure $\nRightarrow$ uniformly: $f_n=n\chi_{[0,\frac{1}{n^2}]}$
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