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!!
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!!
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, $\lim_n{f_n(0)} = 0 \neq f(0)$.