Tail condition (Varadhan's lemma)

Question

I would like your help with the following tail condition, which arises in the theory of large deviations.

Let $P(\mathbb{R}^{d})$ the space of probability measures on $\mathbb{R}^{d}$, $ G:P(\mathbb{R}^{d})\longrightarrow \mathbb{R}$ continuous and bounded, $W:P(\mathbb{R}^{d})\longrightarrow [0,\infty]$, ${Q}$ prob measure on $P(\mathbb{R}^{d})$,

$A=\{-G -\sigma W \geqslant M \}$ a subset of $P(\mathbb{R}^{d})$ and $\sigma$ a positive constant.

Prove that $\lim \limits_{M \to \infty} \limsup \limits_{n \to \infty} \frac{1}{n}\log \int_{A} \exp[n(-G(\mu)- \sigma W(\mu)] d{Q}(\mu) \, = \, -\infty$

So far all I can do is to use that $W\geqslant0$, and bound the quantity inside the limits by $\frac{1}{n}\log \int_{\{-G \geqslant M\}} \exp[n(-G(\mu)] d{Q}(\mu) $, but I am not sure if this is useful at all.

Answer 1

Since $G$ is bounded and $\sigma W\ge0$, for all large enough real $M$ we have $A=\emptyset$, $\int_A\cdots=0$, and $\log\int_A\cdots=-\infty$, which yields what you wanted.