# show that set is measurable

• Nov 10th 2013, 04:48 PM
director
show that set is measurable
Hi all,

Quote:

Given a set E; for each open, bounded interval (a,b):
b-a = m*((a,b)∩E) + m*((a,b)~E)
implies E is (Lebesgue) measurable.

(we're working with the real line here, so E is a subset of R)
I'm having trouble with this problem. Hint given is to show that A = { E: b-a = m*((a,b)∩E) + m*((a,b)~E) } is a sigma algebra.
I understand the hint logic - if A is a sigma algebra then the sets in A are measurable sets so E must be measurable.

I know that all open, bounded intervals are measurable.
I know the three requirements for a collection of sets to be a sigma algebra.

I don't know how to connect the dots though.

Where do I start? Thanks
• Nov 11th 2013, 04:31 PM
director
Re: show that set is measurable
Here is the definition of meausrability of a set E given in my text:

Quote:

A set E is said to be measurable provided for any set A, m*(A) = m*(A∩E) + m*(A∩(complement of E))
It looks like (A∩(complement of E)) = (A~E) or (A\E) (just different notation)

So, the equation in the original problem looks very close to what I have in the definition of measurable set. Except I can't say it holds for any set A, only if A=(a,b).
If I were to restrict the measure space (correct term?) to one that contains the just open sets, then E would be measurable?

The complement of an measurable open set is a measurable closed set... What if I add these in?