TITLE: The Bochner integral about a semigroup of bounded linear operators on a Banach space
QUESTION [5 upvotes]: Let $T(t)$ be a semigroup of bounded linear operators on a Banach space $X$. When does the following hold
$$
\int_0^t T(s)x ds = \Big(\int_0^t T(s) ds\Big)x, \quad x \in X \, ,
$$
where $ t \in (0,1)$?

REPLY [7 votes]: Following ideas of John von Neumann,
J. von Neumann, Über einen Satz von Herrn M. H. Stone, Ann. Math. (2) 33, 567-573 (1932). ZBL0005.16402,
it can be shown that if a function satisfies the semigroup law and is measurable, then it is already continuous. Hence your identity is only true if the generator is bounded.
See also
Engel, Klaus-Jochen; Nagel, Rainer, One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics. 194. Berlin: Springer. xxi, 586 p. (2000). ZBL0952.47036., Exercise I.1.7.(2).