integration – Discard the Pettis integral over inequalities


Let $(E,leq)$ be a Partially Ordered Banach Space.

Let $X:=mathcal{C}(I,E)$ with $I:=(0,1)$.

Let $f:Itimes Erightarrow E$ be a function with $t mapsto f(t,x(t)) $ Pettis integrable
for each $xin X$.

Assume that there exists $x_1,x_2in X$ such that, $forall xin X,;forall tin I$: $$int_{0}^{t}f(s,x_1(s))ds leq int_{0}^{t}f(s,x(s))ds leq int_{0}^{t}f(s,x_2(s))ds.$$

May we discard the integral on the inequalities, and then $forall xin X,;forall tin I$: $$f(t,x_1(t))ds leq f(t,x(t))ds leq f(t,x_2(t))ds.$$

IF NOT, any counterexample is very welcomed.


N.T: I’m not very familiar with Pettis integration that deep, so my question may be not appropriate to MO.