Real analysis: Problem similar to uniformly integrable

1. Nov 23, 2012

happysauce

1. The problem statement, all variables and given/known data

Assume $\mu$(X) >0 and that f is a measurable function that maps X into ℝ and satisfies f(x) >0 for all x$\in$X.

Let $\alpha$ be any fixed real number satisfying 0<$\alpha$<$\mu$(X) <infinity Prove that

inf { $\int$$_{E}$f d$\mu$ : E$\in$M, $\mu$(E) ≥$\alpha$} >0.

(Hint. First prove for any $\delta$ satisfying 0<$\delta$<$\alpha$, prov there exists n $\in$ N such that B$_{n}$ = {x:f(x)$\geq$1/n} satisfies $\mu$(B$_{n}$) $\geq$
$\mu$(X) - $\delta$. Then prove that if $\mu$(E) ≥$\alpha$ then $\mu$(E$\cap$B$_{n}$) ≥$\alpha$-$\delta$

2. Relevant equations

Not sure what is relevant.

3. The attempt at a solution

So, the hint suggests the first thing I do is show that given 0<$\delta$<$\alpha$ i can find an n so that $\mu$(B$_{n}$) $\geq$ $\mu$(X) - $\delta$. So since f(x) >0 then $\mu$(X) = $\mu$({x:f(x)>0}) = $\mu$({x:f(x)≥1/n} $\cup${x:f(x)<1/n}). These are disjoint so we have $\mu$({x:f(x)≥1/n}) + $\mu$({x:f(x)<1/n}). If i show the delta relation with $\mu$({x:f(x)<1/n}) then i get $\mu$(B$_{n}$) $\geq$ $\mu$(X) - $\delta$. It seems really trivial though and im not sure what guarantees this.

The next part asks you to show that $\mu$(E$\cap$B$_{n}$) ≥$\alpha$-$\delta$. Ok so $\mu$(E$\cap$B$_{n}$) = $\mu$(E) +$\mu$(B$_{n}$) - $\mu$(E$\cup$B$_{n}$)

The following hold: $\mu$(E$\cup$B$_{n}$) ≤$\mu$(X) (both are subsets)
$\mu$(B$_{n}$) ≥ $\mu$(X) - $\delta$ (previous part)
$\mu$(E)≥$\alpha$ (hypothesis)

So $\mu$(E$\cap$B$_{n}$) = $\mu$(E) +$\mu$(B$_{n}$) - $\mu$(E$\cup$B$_{n}$) ≥ $\mu$(X) - $\delta$ + $\mu$(E) - $\mu$(X) =$\mu$(E)- $\delta$≥$\alpha$-$\delta$

So I've done most of what the hint wants... the problem is I don't know how this helps me.

2. Nov 23, 2012

happysauce

The title is misleading, I thought it related to uniformly intagrable but I don't think it does.

3. Nov 24, 2012

happysauce

I figured out why there is an n for every δ. So I figured out both parts of the hint, but I still have no clue how to make these hints help me here. Do I have to show that

inf {$\int$$_{B_{n}\cap E}$f d$\mu$, E$\in$M, $\mu (B_{n}\cap E)$≥α-δ} > 0 for each n? And then as n-> infinity we have the desired result? I'm still not sure how to prove this case though...