Inequality in products of probability measures

Question

For each $i=1,\ldots,N$ let $X_i$ be a compact subset of a t.v.s Hausdorff.
Let $M_i$ be the set of regular probability measures on $X_i$ and $u_i: \prod X_i \rightarrow \mathbb{R}$ a measurable bounded function.

If $\lambda=(\lambda_1,\ldots,\lambda_N) \in \prod M_i$ , there exists a $\tilde{x}_i \in X_i$ such that $$ \int_{X_{-i}}u_i(\tilde{x}_i,y)d\lambda_{-i}(y) \geq \int_{X}u_i(x,y)d\lambda_i(x)d\lambda_{-i}(y) $$

Is it true?

Thanks.

Answer 1

Yes. If the inequality is false for every $\tilde {x_i}$ just integrate w.r.t. $\lambda_i$ to get a contradiction.