Let $(\mathcal{X}, \mathcal{F}, (\mathbb{P}_\vartheta)_{\vartheta \in \Theta})$ be a statistical model dominated by a sigma-finite measure $\mu$ with Likelihood-function $L(\vartheta, x)$ which is assumed to be continuously differentiable in $\vartheta$.
One regularity condition, used in the proof of the Rao-Cramer lower bound in our lecture is:
$\int_{\mathcal{X}} \nabla_{\vartheta} L(\vartheta, x) \mu(dx) = \nabla_{\vartheta}\int_{\mathcal{X}} L(\vartheta, x) \mu(dx) \;\;\;\;\;\; (1) $
It was argued that that this condition holds due to the dominated convergence theorem if every $\vartheta_0 \in \Theta$ has a neighborhood $U_{\vartheta_0} \subseteq \Theta$ such that
$\int_{\mathcal{X}} sup_{\vartheta \in U_{\vartheta_0}} \left[\nabla_{\vartheta} L(\vartheta, x) \right] \mu(dx) < \infty\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(2)$
I tried to understand the argument by considering a one-dimensional $\vartheta$ first. If the dominated convergence theorem is to be applied, taking the derivative has to be expressed as the limit of the difference quotient:
$\int_{\mathcal{X}} sup_{n \in \mathbb{N}} \frac{L(\vartheta + \frac{1}{n}, x)-L(\vartheta, x)}{\frac{1}{n}} \mu(dx)<\infty \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(3)$
But i don't understand how i get from here to (a one-dimensional version of) equation (2). Can somebody provide the missing steps?
Edit: Ok i got it. In equation (3) the mean value theorem alows to replace the difference quotient with the derivative w.r.t. $\vartheta$ evaluated at some $\vartheta^*\; \in\; [\vartheta, \vartheta+\frac{1}{n}]$. Since we need to consider an endpiece of the difference-quotient only, $n$ can be chosen arbitrarily large, especially such that $[\vartheta, \vartheta+\frac{1}{n}] \subseteq \Theta$.