Proving Lebesgue Measure on R using Gdelta and Open Sets

[a09]

Homework Statement

[PLAIN]http://www.album.com.hk/d/1546698-2/Untitled-1.jpg

Homework Equations

Every measurable set in R can be written as a difference of Gdelta set and a set of measure zero.
Every open set in R is just a countable union of disjoint intervals.

The Attempt at a Solution

Basically, I have reduced the problem into only concerning Gdelta sets (because sets of measure zero do not really matter that much), then I have further decomposed it into just a problem concerning only of open sets(if assuming it is true for open sets, I think I can prove it for Gdelta sets). So now I am hopelessly stuck, I have been thinking that every open set is just a countable union of disjoint interval, but I have no clue how to prove it for open sets.

 

[a09]

nevermind, I solved it, the solution isn't as complicated as I thought, I do not need to decompose it into Gdelta sets.