Measure Theory-Lebesguq Measure

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Homework Statement


Prove the set A= \bigcup_{n=1}^{\infty} ( \frac{n}{5} , \frac{n}{5} + \frac{n+1}{2^n} ) is Lebesgue measurable and calculate its measure.


Homework Equations


The Attempt at a Solution


I've proved the set is measurable...But how can I calculate its measure?


I will be delighted to get some guidance

Thanks in advance !
 
Last edited:
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It looks to me like
A= (\frac{1}{5}, \infty)
 

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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