Example of differentiating through Lebesgue integral

Question

I am having trouble showing that I can differentiate this specific example through the integral

$$\frac{d}{dt} \int_\mathbb{R} e^{-tx^2} dx = - \int_\mathbb{R} x^2 e^{-tx^2} dx$$

The way I tried was using the Dominated convergence theorem, but I am stuck. I was able to prove that the sequence defined by

$$f_n = \left(\frac{x^2}{n} +1\right)^{-tn}$$

converges to $e^{-tx^2}$ and counts with all the required properties for the theorem. That is, measurability, and that it is absolutely bounded (e.g. by the constant function $1$ in this case). This also applies to the sequence defined by the derivatives $\frac{d f_n}{dt}$.

My problem is that, at the end I have the same problem, the limit is still outside the integral. At this point, I have shown that

$$\int_\mathbb{R} \frac{df_n}{dt} dx \to - \int_\mathbb{R} x^2 e^{-tx^2} dx$$

$$\int_\mathbb{R} f_n dx \to \int_\mathbb{R} e^{-tx^2} dx$$

But still not the crucial fact that

$$\frac{d}{dt} \int_\mathbb{R} f_n dx = \int_\mathbb{R} \frac{d f_n}{dt} dx$$

Answer 1

Let $t_0>0$ and $h_n \to 0$ and $h_n>0$

Then $f_n(x)=\frac{ (e^{-(t_0+h_n)x^2}-e^{-t_0x^2})}{h_n} \to -x^2e^{-t_0x^2}, \forall x \in \Bbb{R}$

Also $$\frac{1}{h_n}[\int e^{-(t_0+h_n)x^2}dx-\int e^{-t_0x^2}dx]=\int\frac{ e^{-(t_0+h_n)x^2}-e^{-t_0x^2}}{h_n}dx=\int f_n(x)dx$$

and $$\int|f_n(x)|dx=\int\frac{ |e^{-(t_0+h_n)x^2}-e^{-t_0x^2}|}{|h_n|}dx$$ $$=\int e^{-t_0x^2} \frac{|e^{-h_nx^2}-1|}{|h_n|}dx$$

Since for $a>0$ the function $e^{-ax^2}$ is Lipschitz we have that exists $M>0$ such that $|e^{-h_nx^2}-1| \leq M|h_n|x^2, \forall n \in \Bbb{N}$

Can you continue from here with the DCT?

For $t \leq 0$ the integrals are infinite.

Answer 2

In general, we can differentiate under the integral $\int_X f(x,t) \, d\mu(x)$ with respect to $t$ if $f(\cdot,t_0)$ is integrable for some $t_0 \in [a,b]$ and there exists an integrable function $x \mapsto g(x)$ such that $\left|\frac{\partial}{\partial t} f(x,t)\right| \leqslant g(x)$ for all $t \in [a,b]$.

This can be proved with a combination of the mean value theorem and dominated convergence theorem. What is interesting about this theorem is you only need to assume integrability at a single point $t_0$ and the rest follows from the bound on the partial derivative.

Here we have, assuming $0 < a < b$,

$$\left|-x^2e^{-tx^2}\right| \leqslant g(x) =x^2 e^{-ax^2}, $$