Show that on space $([0,1],\lambda,\mathcal{L})$ where $\lambda$ is Lebesgue measure defined on Lebesgue $\sigma-$ algebra $\mathcal{L}$ .
function $f_n(x) =x^n$ on $[0,1]$ is not uniformly converge to $f =0$ almost everywhere.
It's easy to see that $x^n$ converge uniformly to $0$ on $[0,a]$ for any fixed $a<1$.
But I can't figure out why there is not any zero measure set $E^c$such that on its complement $f_n(x)$ converge uniformly to $0$.
A related question:A sequence of continuous functions on [0,1] which converge pointwise a.e. but does not converge uniformly on any interval
Best Answer
Suppose $E^{c}$ has measure $0$ and $x^{n} \to 0$ uniformly on $E$. Then $E$ is dense. Hence there exists a point $x_n \in (1-\frac 1 n, 1) \cap E$. Now $sup \{x^{n}: x \in E\} \geq x_n^{n} >(1-\frac 1n)^{n} \to \frac 1 e$, a contradiction.