Hello

I know when the limits of integration are finite its possible to bring the derivative sign inside the integral (i.e. the generalized Leibniz integral rule). In its basic form the rule says

$\displaystyle \frac{d}{\text{dx}}\int _{y_0}^{y_1}f(x,y)dy=\int _{y_0}^{y_1}\frac{\partial }{\partial x}f(x,y)dy$.

But what if $\displaystyle y_0 = -\infty $ and $\displaystyle y_1 = + \infty $?

I've seen people bring the derivative sign inside the integral in this case, but since the limits are not finite the Leibniz rule does not apply directly. So how can this operation be performed. What theorem applies??

I have a suspicion it may have something to do with Lebesgues Dominated Convergence Theorem.