# 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.