Measure Theory Problem

Hello all,


I have a few questions in my mind:

1) [tex] \lim_{n\rightarrow \infty}[0,n) = \cup_{n\in\mathbb{N}}[0,n) = [0,infty) [/tex] holds, and for [tex] \lim_{n\rightarrow \infty}[0,n] = \cup_{n\in\mathbb{N}}[0,n] = [0,infty) [/tex] is also true? It should not be [tex] [0,infty] [/tex], am I correct?

2) Consider an extended real function [tex] f [/tex], if we use simple function [tex] f_{n} = f 1_{f\leq n} [/tex], by taking limit, we can only have it approximated to [tex] f 1_{f < \infty} [/tex] but since [tex] f [/tex] may take [tex] \infty [/tex], such simple function may not be approximating [tex] f [/tex] almost everywhere unless [tex] f = \infty [/tex] is of measure 0?

Am I correct? Thanks.


Wayne
 

mathman

Science Advisor
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Your first statement is correct. I can't figure out the second question - the symbol 1 after f in the expression fn= means what?
 

Landau

Science Advisor
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Help me out here. What does
[tex] \lim_{n\rightarrow \infty}[0,n)[/tex]
mean? Or were you just defining it as
[tex] \lim_{n\rightarrow \infty}[0,n) := \cup_{n\in\mathbb{N}}[0,n)\?[/tex]
 
the symbol 1 here means the indicator function.

Thanks.
 

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