In my course on discrete-time finance we derived the following equality for a lower bound for the value of a not necessarily replicable contingent claim $D$. Here we are looking at a single period market with risk-free interest rate $r$ and $E$ is the expectation operator. $$\sup\{E[Y(1)/(1+r)]:Y \text{ replicable, }\text Y\le D\} \\ = \inf\{E_{Q'}[D(1)/(1+r))]:Q' \text{ risk free P-measure}\text \}$$ My question: Which probability measure do we use to compute the first expectation on the left? Is it a risk-neutral one, or is it the real one?
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Coming back to this question way later, I managed to resolve my confusion. Using a well-known theorem we have that $E_{Q}[Y(1)]$ is constant on $\mathbb{M}$. Thus, it is irrelevant which measure we are using, since the resulting bound will be the same.