Let us consider the following integral: $$I(x)=\int_\Omega f(x, \omega)\, d\omega, $$ where $\Omega$ is a measure space and $f\colon \mathbb{R}\times \Omega \to \mathbb{R}$ is such that $f(x, \cdot)\in L^1(\Omega)$ for all $x$.
When can we differentiate $I$? A dominated convergence argument gives the following result.
Proposition. If
- For almost all $\omega\in \mathbb{\Omega}$, $$f(\cdot, \omega)\ \text{is differentiable;}$$
- For all $x\in \mathbb{R}$, $$\frac{\partial f}{\partial x}(x, \cdot)\in L^1(\Omega);$$
- There exists a function $\Theta\in L^1(\Omega)$ such that $$\left\lvert \frac{\partial f}{\partial x}(x, \omega)\right\rvert \le \Theta (\omega).$$
Then $I$ is differentiable and $$\frac{d I}{dx}=\int_\Omega\frac{\partial f}{\partial x}(x, \omega)\, d\omega.$$
(This has appeared here sometimes, for instance in this answer by Qiaochu Yuan.)
My question is:
Can you provide an example showing that condition 3 cannot be dropped?
