Showing that the hypothesis that $m (E) <\infty $ is essential in the Egoroff's theorem.

Question

In Egoroff theorem, the hypothesis that $ m (E) <\infty $ is essential. Construct an example of measurable functions $ f_n: \mathbb{R} \rightarrow \mathbb {R} $ that converge to the null function with the following property: if $ F \subset \mathbb{R} $ and $ m (R \backslash F) <\infty $ then $ \{f_n \}$ not converges uniformly on $ F $

I have not idea how to do this! I'm thinking for days but to no avail! I'm terrible with examples ...

Answer 1

Let $f_n:\mathbb{R}\rightarrow\mathbb{R}$ where $$ f_n(x)= \left \{ \begin{array}{ll} 1 & \textrm{ if } x\in[n,\infty) \\ 0 & \textrm{ if } x\not\in[n,\infty) \end{array} \right. $$ Let $F\subset \mathbb{R}$, $m(\mathbb{R}\backslash F)<\infty$ a closed set.

For each $n\in \mathbb{N}$ exists $x\in F$ and $N\geq n$ such that $f_N(x)=1$. This works because F is very large and always lets us find these elements (in $[N,\infty)$)