Let $E$ be a $\mathbb R$-Banach space and $(T(t))_{t\ge0}$ be a semigroup on $E$. Assume $$[0,\infty)\to E\;,\;\;\;t\mapsto T(t)x\tag1$$ is Borel measurable for all $x\in E$ and $$\forall t>0:\sup_{s\in[0,\:t)}\left\|T(t)\right\|_{\mathfrak L(E)}<\infty\tag2.$$
Let $t\ge0$ and $x\in E$. Are we able to show that $$\frac1h\int_t^{t+h}T(s)x\:{\rm d}s\xrightarrow{h\to0+}T(t)x?\tag3$$
If the orbits $(1)$ are continuous, then the left-hand side of $(3)$ is an ordinary Riemann integral and hence $(3)$ follows from the fundamental theorem of calculus. How can we show it in general (when the integral in $(3)$ is a Lebesgue integral)?