Let $(\Omega,\mathcal{F})$ denote some measurable space. Let $P_1$ and $P_2$ denote respectively two probability measures. Now let $\mathcal{G}$ be some sub sigma-algebra of $\mathcal{F}$. Given a positive integrable random variable $X$, we can define respectively the conditional expectation

$$Y_1=E^{P_1}[X|\mathcal{G}],~ Y_2=E^{P_2}[X|\mathcal{G}]$$

Now for some $0<\alpha<1$, we can define a new probability measure $P=\alpha P_1+(1-\alpha)P_2$, then we get

$$Y=E^{P}[X|\mathcal{G}]$$

Now my question is whether we can prove

$$\operatorname{esssup}{}_P(Y)\le \alpha \operatorname{esssup}_{P_1}(Y_1)+(1-\alpha)\operatorname{esssup}_{P_2}(Y_2)?$$

Here the definition of $\operatorname{esssup}_{Q}(\cdot)$ w.r.t some probability $Q$ can be found here:

http://en.wikipedia.org/wiki/Essential_supremum_and_essential_infimum

Thanks a lot for the help!