Im at the beginning of my studies of operator semigroup theory and I have some trouble understanding the integration of operators in Banach Spaces.
Let $(T(t))_{t\geq0}$ be a $C_0$-semigroup on a Banach space $X$ with norm $|| \cdot ||$.
My problem is that in general I see that the Bochner integral is the standard integral for dealing with Banach space valued functions but often authors use the fact that \begin{equation} \lim_{h\downarrow 0} \frac{1}{h}\int_{t}^{t+h} T(s)x\,ds=T(t)x \end{equation} by the strong continuity of the semigroup $(T(t))_{t\geq0}$ and the Fundamental Theorem of Calculus for the Rieman integral for Banach spaces.
I have knowledge of measure theory for real valued functions and I know that there is a connection between the Lebesgue and the Riemann integral for real valued functions $f$, namely:
For a bounded intervall $[a,b]\subset \mathbb{R}$ every Riemann integrable function $f$ is also Lebesgue integrable and \begin{equation} \int_{[a,b]}f\,d\lambda=\int_a^{b}f(x)\,dx, \end{equation} where $\lambda$ denotes the lebsegue measure on $\mathbb{R}$.
Is there a similar result which connects the Bochner integral and the Riemann integral for functions with values in Banach spaces?
I don't have a general answer because I'm not sure if there is a characterization of Riemann integrability for Banach-space-valued functions. But you can easily see that the Riemann sums can be seen as Bochner integrals of simple functions. Then if $f$ is continuous, for instance, as your domain is compact these simple functions will converge uniformly to $f$, and so the Bochner integral will be precisely the Riemann integral.