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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 $\frac{a+b}{2}$.2012-09-08

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