Observe that the differentiability property of Theorem 7.2 conversely implies the orthogonality property. Indeed, pick $f\in C^{0}(X)$ and set $g:=P_{\omega}(f)-f$. We claim that $\int g\MA(P_{\omega}(f))=0$. It is enough to prove $\int g\MA(P_{\omega}(f))\geq 0$ since $g\leq 0$. Now the differentiability property yields

$$E_{\omega}(P_{\omega}(f+\varepsilon g))=E_{\omega}(P_{\omega}(f))+\varepsilon\int g\MA(P_{\omega}(f))+o(\varepsilon).$$

But we have

$$f+\varepsilon g=(1-\varepsilon)f+\varepsilon P_{\omega}(f)\geq(1-\varepsilon)P_{\omega}(f)+\varepsilon P_{\omega}(f)=P_{\omega}(f),$$

hence $E_{\omega}\left(P_{\omega}(f+\varepsilon g)\right)\geq E_{\omega}(P_{\omega}(f))$ by monotonicity of $E_{\omega}$, and the result follows.