# Convergence in measure and pointwise convergence in continuity points

Hi can you help me with the following:

$\{f_n\}$ a sequence of increasing functions with $f_n\to f$ in measure on $[a,b]$. Show that $f_n(x)\to f(x)$ at every $x$ where $f(x)$ is continuous.

Thanks a lot!!

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Do you mean the sequence is increasing? Or do you mean you have a sequence of strictly increasing functions? The first case I think is false. Consider $f = 3$, and $f_n = 3$ everywhere except at $\frac{a+b}{2}$ and $\frac{a+2b}{2}$. At $\frac{a+b}{2}$, $f_n = 0$. At $\frac{a+2b}{2}$, $f_n = 3 - \frac{1}{n}$. This is an increasing sequence of functions that certainly converges in measure but $f_n(\frac{a+b}{2}) \not\to f(\frac{a+b}{2})$. However, $f$ is clearly continuous at $\frac{a+b}{2}$. – Shankara Pailoor Sep 8 '12 at 18:45

I believe that even if you say the sequence consists of non-decreasing functions, the statement is false. Let $f(x) = 3$ on $[a,b]$. Take, $$\{f_n\} = \left\{ \begin{array}{lr} 3 & \text{if } x \neq a\\ 0 & \text{if } x = a \end{array} \right.$$
$f_n$ is clearly non-decreasing and converges to $f$ in measure. However, $\lim_n{f_n(0)} = 0 \neq f(0)$.
you are right, we can just show for a particular $n$ that $|f_n(x) - f_n(z)| < \epsilon$. Sorry about that. – Shankara Pailoor Sep 8 '12 at 21:04
One more question...is there no condition that says $f_n(x) \leq f_{n+1}(x)$? – Shankara Pailoor Sep 8 '12 at 23:05