Equivalence in the formulation of the "Large Deviation Principle"

Question

I'm reading this notes on the Internet http://staff.utia.cas.cz/swart/lecture_notes/LDP4.pdf. And I'm stuck in this proposition:

Let E be a Polish space. A sequence of finites measures $\{\mu_n\}$ with a good rate function $I:E\longrightarrow[-\infty,\infty]$ satisfies that $$\lim_{n\longrightarrow\infty}\left(\int f^n\mathrm{d}\mu_n\right)^{\frac{1}{n}}=\sup_{x\in E}f(x)e^{-I(x)},\qquad\forall f\in C(E), f\geq 0$$ if and only if the following conditions hold:

a) $\displaystyle\limsup_{n\longrightarrow\infty}\frac{1}{n}\log \mu_n(C)\leq -\inf_{x\in C}I(x),\quad \forall C\subseteq E \:\mathrm{closed}$

b)$\displaystyle\liminf_{n\longrightarrow\infty}\frac{1}{n}\log \mu_n(G)\geq -\inf_{x\in G}I(x),\quad \forall G\subseteq E \:\mathrm{open}$

What I've done so far:

I was able to prove the necessity condition. But I'm stuck in the proof of the sufficiency condition.

My attempt: Let C be a closed subset of E. Then we have that $$\displaystyle\limsup_{n\longrightarrow\infty}\frac{1}{n}\log \mu_n(C)\leq -\inf_{x\in C}I(x)$$ $$\Leftrightarrow \displaystyle\limsup_{n\longrightarrow\infty}\log \mu_n(C)^{\frac{1}{n}}\leq \sup_{x\in C}-I(x)$$
Using the monotonicity and the continuity of the exponential we have that $$\Rightarrow \displaystyle\limsup_{n\longrightarrow\infty} \mu_n(C)^{\frac{1}{n}}\leq \sup_{x\in C}e^{-I(x)}$$ $$\Leftrightarrow \displaystyle\limsup_{n\longrightarrow\infty} \left(\int_E\chi_C^n\mathrm{d}\mu_n\right)^{\frac{1}{n}}\leq \sup_{x\in C}e^{-I(x)}$$ So I don't know how to proceed to get $$ \displaystyle\limsup_{n\longrightarrow\infty} \left(\int_E\chi_C^n\mathrm{d}\mu_n\right)^{\frac{1}{n}}\leq \sup_{x\in C}e^{-I(x)}\chi_C(x)$$ Did I do a wrong step when I handled the inequalities?

Answer 1

Notice that $\sup_{x\in C}e^{-I(x)}=\sup_{x\in C}e^{-I(x)}\chi_C(x)$. Indeed, $e^{-I(x)}=e^{-I(x)}\chi_C(x)$ for all $x\in C$. However, I'm not sure whether this approach will prove fruitful - the proof in the notes you linked uses several lemmas over multiple pages.

If this is your first time learning about large deviations principles, I would suggest using a different source. These notes use very nonstandard definitions which, in my opinion, remove most of the intuition behind LDPs. While everything is equivalent (as shown in the very proposition you are trying to prove) and there are certainly benefits to this construction, it is less accessible to a newcomer and will make things harder to understand when you use other sources in the future.

A good alternative is this set of notes by Rezakhanlou. It defines LDPs in the usual way and then proves Varadhan's lemma, which is essentially equivalent to your proposition.