When we value the floating leg of a standard vanilla swap, we replace the expectation of the future floating rates by the forward rates known today. However my understanding is that the forward rate is equal to the expectation of a spot rate only under the corresponding maturity forward measure. So for example if we have a spot rate r(T,D) that is known at time T but paid at time T+D then the forward rate f(T,D) known at time 0 is equal to the expectation of r(T,D) only under the T+D forward measure.
Now I don't understand why for vanilla swaps, we are not concerned about that when we replace the future libor spot rates by their forward rates? Or are we implicitly assuming that each payment of the floating leg is priced under its own forward T_i+D measure for example (where T_i is the time we observe the Libor rate and T_i+D is the time we pay it)?