Proving Lebesgue Measure on R using Gdelta and Open Sets

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

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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