(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove the following: If $\{f_n\}$ is a collection of measurable functions defined on $\rr$ and satisfying:

1) $f_n(x)\le 1 \, \forall n \in \nn \, \forall x \in \rr$ and

2) $f_n(x) \ge 0 \mbox{ a.e. on } \rr \, \forall n \in \nn$ and

3) $\ds f(x)=\sup\{f_n(x) \mid n \in \nn\}$,

then $f(x) \ge 0$ a.e. on $\rr$.

2. Relevant equations

Definition of 'almost everywhere,' knowledge of measurability, supremums etc.

3. The attempt at a solution

I've been racking my brain trying to figure out..what's going on in this problem - I'm not too good at comprehending 'almost everywhere.' I think I want to show that \{ x \in \rr | f(x) > 0\} has measure 0, but I don't really know how to get there. I began by substituting, so I'd be showing \{x \in \rr | \sup\{f_n(x) | n \in \nn\} > 0\} has measure 0, but that seems too messy and convoluted to be correct.

I think this is a relatively straight-forward proof, but I'm not entirely sure how to get started. A gentle push in the right direction would be great.

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# Homework Help: 'Almost everywhere' proof

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