It is well known that [itex]\sin x / x[/itex] is not Lebesgue integrable on [itex][0, +\infty)[/itex] though it is (improper) Riemann Integrable. It is also fairly easily shown (integrating by parts) that(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\Bigg\lvert \int\limits_{a}^{b} \frac{\sin x}{x} dx\Bigg\rvert \leq 4[/tex]

Since [itex][a,b][/itex] is compact, the Riemann and Lebesgue Integral of [itex]\sin x / x[/itex] coincide on this set. As [itex]b \to \infty[/itex] and [itex]a\to 0[/itex] the upper bound of 4 remains valid, though in the limit, the Lebesgue integral does not exist.

I am reading a book that asks the reader to prove the above bound, and in the text, it uses this fact in the computation of this integral:

[tex] \lim_{\epsilon \to 0}\int\limits_{-\infty}^{+\infty}f(y) \Bigg\lvert \int\limits_{\epsilon^{-1}\geq x \geq \epsilon} \frac{\sin xy}{x} dx \Bigg\rvert dy [/tex]

If we let [itex]F_{k}(y) = f(y) \Bigg\lvert \int\limits_{k^{-1}\geq x \geq k} \frac{\sin xy}{x} dx \Bigg\rvert [/itex] and the bound of 4 above, we dominate [itex]F_k(y)[/itex] by [itex]f(y)4[/itex].

If we assume [itex]f[/itex] is bounded and in [itex]L^1[/itex] then we can use Lebesgue Dominated Convergence Theorem to pass the limit inside the integral.

Now, I get that [itex]F_k[/itex] is uniformly bounded in [itex]k[/itex]. However, if the integral is taken to be a Lebesgue integral (which, it is initially) then I don't see how [itex]\lim F_k [/itex] is even defined. So, what is going on?

So, perhaps I am missing something obvious, but I am just a little confused by this. If I have missed something completely obvious, please don't hesitate to tell me!

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Integral of sinx/x

**Physics Forums | Science Articles, Homework Help, Discussion**