Real Analysis – Using Leibniz Integral Rule on Infinite Region

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I am trying to take the derivative with respect to $a$ of some function $I(a)=\int_{0}^{\infty}f(a,x)dx$. I would like to make sure that I am using the Leiniz Integral Rule correctly. Various web sources indicate a set of conditions that must hold for $f(x,a)$ and $\frac{\partial f(x,a)}{\partial a}$ when integration is done over infinite region. From reading this source (see Theorem 10.3 on page 13) the conditions that $f(x,a)$ and $\frac{\partial f(x,a)}{\partial a}$ must obey are:

  1. $f(x,a)$ and $\frac{\partial f(x,a)}{\partial a}$ are continuous over $x\in[0,\infty)$ and around $a$ that we are interested in.

  2. There exists an integrable function (over $x$) $g(x)$ such that $|\frac{\partial f(x,a)}{\partial a}|\leq g(x)$.

  3. There exists an integrable function (over $x$) $h(x)$ such that $|f(x,a)|\leq h(x)$.

Integrable here means $\int_{-\infty}^{\infty}g(x)dx<\infty$.

However, another source seems to omit condition 3 above. I am wondering which source is correct. If there are "both correct", when is condition 3 necessary?

Best Answer

EDIT:

Revisiting this question, I now realize that the author of the paper is using the uniform convergence of the integral, not the dominated convergence theorem, to justify differentiating under the integral sign.

A proof can be found in the first supplement to the textbook Introduction to Real Analysis by William F. Trench. It's Theorem 11 on page 18.

Condition 2 is the M-test for uniform convergence, which can only be used if $\int^{\infty}_{0} \frac{\partial}{\partial a} f(x,a) \, dx $ is absolutely convergent.

The uniform converge of $\int_{0}^{\infty} f(x,a) \, dx$ is not one of the assumptions in the proof, but rather it's proven that it's a consequence of $\int^{\infty}_{0} \frac{\partial}{\partial a} f(x,a) \, dx $ converging uniformly for values of $a$ in some closed interval.

Those are sufficient but not necessary conditions. Basically they mimic Theorem 2 at the following link without any reference to measure theory or Lebesgue integration. Condition 3 from your source is basically saying that $f(x,a)$ is Lebesgue integrable.

http://planetmath.org/differentiationundertheintegralsign

I prefer the related approach (when possible) of expressing the integral as an iterated integral and then switching the order of integration. Justifying switching the order of integration is usually easier.

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