1. ## 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.

2. 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 { $I^n_k$} suvh that $\sum_{k=1} {m(I^n_k)} \leq m(E) + 1/n$

Let $G_n = \bigcup_{k=1}{I^n_k}$. Then $G_n$ is borel since it is the union of open sets.

Let $G = \cap{G_n}$, 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, $m(G) \leq m(G_n) \leq m(E) + 1/n$

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

and m(G/E) = 0