Continuous limit of continuous functions with a distance condition Let $\{f_n\}_n$ be a sequence of continuous functions $f_n\colon[0,1]\to\Bbb R$ and let $U_n=\{\,x\in[0,1]\mid f_n(x)>1\,\}$.
We know that $\forall x\in[0,1]\colon \lim_{n\to\infty}f_n(x)=0$ does not imply that $f_n\to 0$ uniformly. However, the usual counterexamples all have $\lim_{n\to\infty}\mu(U_n)=0$. Is there a counterexample where $\mu(U_n)\ge p>0$ for all $n$? And if so, for which $p$ (certainly not for $p=1$)?
Intuitively, my answer is "no", because it feels as if for "random" $x\in[0,1]$, the "probability" that $f_n(x)\ge1$ should be at least $p$, contradicting $f_n(x)\to 0$.

In fact it can be generalisable, for $\epsilon > 0$ let $U_n(\epsilon) = \{x \in [0,1] : f_n (x) \ge \epsilon\}$ and let $X$ a random variable uniformly distributed on $[0,1]$. Since $f_n \to 0$ pointwise then $f_n(X) \to 0 \text{ a.s.}$ then $f_n(X) \to 0$ in probability. and so $\mu (U_n(\epsilon))=\mathbb P(f_n (X) \ge \epsilon ) \to 0$
