Proving E is Measurable with Compact Sets

[jdcasey9]

Homework Statement

Prove that E is measurable if and only if E $$\bigcap$$ K is measurable for every compact set K.

Homework Equations

E is measurable if for each $$\epsilon$$ < 0 we can find a closed set F and an open set G with F $$\subset$$ E $$\subset$$ G such that m*(G\F) < $$\epsilon$$.

Corollary 16.17: Open sets, and hence also closed sets, are measurable.

Lemma 16.14: If E₁ and E₂ are measureable sets, then so are E₁$$\bigcup$$E₂, E₁$$\bigcap$$E₂ and E₁\E₂.

The Attempt at a Solution

Assume E is measurable. Since a compact set is always closed, K is closed and is measurable by Corollary 16.17 (Carothers). Then, because E and K are both measurable, E$$\bigcap$$ K is measurable by Lemma 16.14 (Carothers).

Assume E$$\bigcap$$K is measurable for compact K. Since K is compact, it is necessarily closed. So E$$\bigcap$$K is also compact and closed because a closed subset of a compact set is compact. Then, because E$$\bigcap$$K is measurable, it is between a closed set F and an open set G. Since it is closed, it must be equal to F for m*(G\F) < $$\epsilon$$.

For all integers n, F_n (closed set), G_n )(open set) with F_n$$\subset$$ E$$\bigcap$$K$$\subset$$G_n such that m*(G_n\F_n) < 2^-n$$\epsilon$$ where G = $$\bigcup$$ G_n is an open set containing E and E=F=$$\bigcup$$F_n is a closed set. So, G\F $$\subset$$ $$\bigcup$$ (G_n\F_n) and m*(G\F) $$\leq$$ $$\Sigma$$ m*(G_n\F_n) < $$\Sigma$$ 2^-n$$\epsilon$$ = 3$$\epsilon$$. So E is measurable.

Is this a sound argument?

 

[micromass]

Hello jdcasey09!

I'm afraid there is a flaw in the argument:

Assume E$$\bigcap$$K is measurable for compact K. Since K is compact, it is necessarily closed.

This is not true. $$E\cap K$$ is simply a measurable subset of K, it can be everything. thus it is not necessarily closed!

For example, if K=[-2,2] and E=]-1,1[, then $$E\cap K$$ is a measurable subset of K, but it isn't closed.

I also see that you don't specify what K is in your proof. That is, you keep K general. This is good, but eventually you'll need to give a specific form for K...