# Measurable Functions

• Sep 25th 2011, 10:00 AM
Markeur
Measurable Functions
Let $(f_n)$ be a sequence of real-valued measurable functions on $[a,b]$. If $(f_n)$ converges to a function $f$ almost everywhere on $[a,b]$, prove that there exists a sequence $(E_n)$ of measurable subsets of $[a,b]$ such that
$m([a,b] \setminus \cup E_n) = 0$
and $(f_n)$ converges uniformly to $f$ on each $E_n$.

How do I even start this question?

• Sep 25th 2011, 02:25 PM
SlipEternal
Re: Measurable Functions
If $(f_n)$ converges almost everywhere to $f$, then there is a zero set of points on which it does not converge. A zero set has certain properties. For one thing, it has measure zero. So, if $$a,b$ \setminus \cup E_n = Z\text{ where }Z\text{ is your zero set}$, then you are guaranteed that the measure will be zero. So, let's construct $E_1$. Since the zero set is a subset of $$a,b$$, by the well ordering principle of real numbers, there exists a greatest lower bound of $Z$. Can you find a really small interval $E_1$ that will include $a$, but not $a_1$? Will that interval be measurable? How do you know? Will $(f_n)$ converge to $f$ uniformly on that interval? Again, how do you know?
• Sep 25th 2011, 09:38 PM
Markeur
Re: Measurable Functions
Is a_1 a point in the set Z or just part of the partition of [0,1]?
• Sep 26th 2011, 02:45 AM
SlipEternal
Re: Measurable Functions
It is a point in the set Z.
• Sep 27th 2011, 08:50 AM
Markeur
Re: Measurable Functions
Alright I managed to solve it. It's actually applying Egorov's Theorem to each E_n and intersect [a,b]\E_n for all n to get m([a,b]\(\cup E_n)) = 0 and f_n converges uniformly to f on each E_n.

Thanks.