Time integrals of semigroup orbits lie in the generator domain

The orbit $s\mapsto T(s)x$ is continuous on the compact interval $[0,t]$ by [F2], hence Bochner integrable there by [F3]; therefore $J_{t}x:={\int }_0^{t}T(s)xds$ is a well-defined element of $X$.

By [F6] applied to $T(h)$ and [F2], $T(h)J_{t}x={\int }_0^{t}T(h)T(s)xds={\int }_0^{t}T(s+h)xds$.

Shift identity for continuous integrands: if $g$ is continuous on $[0,t+h]$, then ${\int }_0^{t}g(s+h)ds={\int }_h^{t+h}g(u)du$. For an integrable simple function $s={\sum }_j{c}_j{\mathbf{1}}_{E_j}$ the identity holds termwise, since translating $E_j\cap [h,t+h]$ back by $h$ gives $E_j-h$ intersected with $[0,t]$, a set of the same measure by [F7]; for a nonnegative measurable function it follows by taking the supremum of the pairings of dominated simple functions, and for a Bochner integrable $g$ it follows by applying the scalar case to the nonnegative integrable $\mathrm{\parallel }g-s_n\mathrm{\parallel }$ and the simple case to $s_n$ along a defining approximation with ${\int }_h^{t+h}\mathrm{\parallel }g-s_n\mathrm{\parallel }\to 0$ [F5]. A continuous $g$ on the compact interval is Bochner integrable by [F3].

Adding the identity ${\int }_0^{t+h}={\int }_0^h+{\int }_h^{t+h}={\int }_0^t+{\int }_t^{t+h}$ of [F4], [steps 1.2 and 1.3] give $(T(h)-I)J_{t}x={\int }_h^{t+h}T(u)xdu-{\int }_0^{t}T(u)xdu={\int }_t^{t+h}T(u)xdu-{\int }_0^{h}T(u)xdu$.

Dividing by $h>0$ and applying the average-convergence limits of [F3] to the continuous orbit at the points $t$ and $0$ (where $T(0)x=x$) yields $\frac{T(h)J_{t}x-J_{t}x}{h}=\frac{1}{h}{\int }_t^{t+h}T(u)xdu-\frac{1}{h}{\int }_0^{h}T(u)xdu⟶T(t)x-x$.

By the definition of the generator [F1], the convergence of these right difference quotients means exactly that $J_{t}x\in D(A)$ and $A{J}_{t}x=T(t)x-x$.

Since $x\in X$ and $t\ge 0$ were arbitrary, for every $x$ and every $t$ the integral $J_{t}x$ lies in $D(A)$ and $A{\int }_0^{t}T(s)xds=T(t)x-x$; at $t=0$ both sides are $0$.