Context first: I'm brushing up a little on measure theory, mainly trying to get a better understanding of integration with respect to measures other than the Lebesgue measure.
I came across the following in a research article:
Let $X$ be a Banach space, $\eta: \mathbb{R}_-\to \mathcal{L}(X)$ be of bounded variation such that $|\eta|(\mathbb{R}_-)<\infty,$ where $|\eta|$ is the positive Borel measure in $\mathbb{R}_-$ defined by the total variation on $\eta$ and $\Phi : C_0(\mathbb{R}_-,X) \cap L^p(\mathbb{R}_-,X) \to X$ be the map $$f \mapsto\int_{-\infty}^0 f \ d\eta.$$
I want to practice computing $\Phi$ for different $f.$ However, due to my lack of knowledge, I am not sure what examples of $\eta$ fit the situation. So what I'm looking for is reference/examples of such $\eta$ (the simpler the better). I have no qualms about sticking to the particular case $X = \mathbb{C}$ for simplicity since I just want to work on computing $\Phi$ in concrete cases.
Best Answer
I suggest you stick to $X=\mathbb C$ if you want to compute anything explicitly.
Here are some cases you should try:
$\eta$ is a continuous function
$\eta$ is a simple function