Lebesgue integral

[katreenaa]

Dear all,

Actually, I have question in the attached file about lebesgue integral & I hope to help me please.

thanks in advance,​

 

[JSuarez]

$$\frac{1}{1-e^{bx}}=\sum_{n=0}^{\infty}e^{nbx}$$

This will allow you to calculate the integral, in a few properly justified steps.

 

[katreenaa]

Can you explain more please JSuarez?

 

[JSuarez]

The partial sums of the above series converges monotonically to the function on the right. If you substitute in the in the integral, what do the Lebesgue's convergence theorems tell you about exchanging it with the integral? If that is permissible, you'll have a series with terms that are almost immediate integrals.