A Property of set with finite measure

[ntsivanidis]

Homework Statement

If E has finite measure and $$\epsilon$$>0, then E is the disjoint union of a finite number of measurable sets, each of which has measure at most $$\epsilon$$.

Homework Equations

The Attempt at a Solution

I proceeded by showing that by definition of measure, there is a finite group of open sets $$O_i$$ that contain E, whose union has the same measure (and contains E). By taking their closure, by compactness each has an open cover of $$\epsilon$$ neighborhoods of a finite number of points. The union of these, within each $$O_i$$ and then across all $$O_i$$, contains E.

My problem is i)to ensure the finite number of subsets are disjoint, and ii) to ensure that the union of these sets is equal to E.

Thanks!

 

[arkajad]

Then why don't you take the intersections of your sets with E and then differences and intersections of the sets themselves?