Thread: borel set

Show that every lebesgue measurable subset of R is the union of a Borel subset of R and a set of of outer measure zero.

Originally Posted by Chandru1

Show that every lebesgue measurable subset of R is the union of a Borel subset of R and a set of of outer measure zero.

Let E be a lebesgue measurable subset of R

For n = 1,2,3,... we can find a set of open intervals { } suvh that

Let . Then is borel since it is the union of open sets.

Let , then G is also Borel.

Finally, I claim m(G\E) = 0, it suffices to show m(E) = m(G)

Since E is in G, then m(E) <= m(G)

And,

And this is true for all n, thus m(G) <= m(E), and so m(E) = m(G)

and m(G/E) = 0