For a specific probability density function $f$ with support on ${\mathbb R}$, which is not differentiable everywhere, I have proven that the Hessian matrix of $$g(\theta) = \int \log f(x;\theta)d H(x),$$ exists for all $\theta \in {\mathbb R}^p$, where $H$ is another distribution function. Let $F(x;\theta) = \int_{-\infty}^x f(t;\theta)dt$ be the distribution function. I want to check if the Hessian of $$G(\theta) = \int \log F(x;\theta)dH(x),$$ also exists.

Is there a direct method of showing this? This is, some general result I can appeal to? If it wasn't for the logarithm, I could use exchange the integral and derivative symbols, for instance.