(This is an associated question to Scaling in utility maximisation. $c_t$, $w_t$, $n_t$, $A$ are defined there.)
I am reading that because of time homogeneity $$\sup_{(n,c)\in A(w)}E\left(\int_t^\infty e^{-\rho t}u(c_t)dt|w_t=w\right)=e^{-\rho t}\sup_{(n,c)\in A(w)}E\left(\int_0^\infty e^{-\rho t}u(c_t)dt\right).$$ I am thinking that time homogeneity means problem does not depend on time but on the interval ie for same periods of time, the problem is the same. How is this being used here?