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

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 ...

• $f_n=\chi_{[n,n+1]}$. Feb 3 '14 at 14:33
• @DavidMitra I will try to show that it satisfies all of these properties of the exercise. It is not obvious to me. Feb 3 '14 at 14:37
• Even easier, take $f_n=\chi_{[n,\infty)}$. Feb 3 '14 at 14:42
• @DavidMitra But... you do not need to fix a set $F$ such that all works? Feb 3 '14 at 15:06
• Yes. Fix $F$ and show that for any $n$ there is an $x\in F$ and $N\ge n$ with $f_N(x)=1$. This will show that $(f_n)$ does not converge uniformly to $0$ on $F$. Feb 3 '14 at 15:09

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 $[N,\infty)$)