2

I'm trying to understand why $\frac{\sin(x)}{x}$ is not integrable according to Lebesgue over $\mathbb{R}$

If found this answer that helped me a lot but don't understand how the Monotone Convergence Theorem was used

https://math.stackexchange.com/a/3184570/741674

Can someone help me by detailing the problematic line in this answer?

Thank you

Keen-ameteur
  • 7,663
jp.perk
  • 21
  • 2
  • It's possibly worth noting that you can sidestep the issue and just notice that over the intervals $[2\pi + \pi/6, 2\pi + 5\pi/6]$ this function is bounded below by $\frac{1}{2x}$, which allows a more elementary argument (even to the level of integral simple functions, if one desires!) based on the divergence of the harmonic series. – Milo Brandt Jan 14 '20 at 15:48

1 Answers1

3

The author is applying the MCT to the non-decreasing sequence of functions $$f_N(x)=\sum_{k=0}^N\Bbb{1}_{[{2k\pi},\, {(2k+1)\pi}]}(x)\frac{\sin(x)}x $$ where $$\Bbb 1_S(x)=\cases{1 & if $x \in S$,\\0 & otherwise.}$$

Mars Plastic
  • 4,239
Arnaud Mortier
  • 27,276