Limit laws how to write an integral as a max of a sum

Question

Good Morning, I am not able to prove following equation:

lim 1/n log()=... https://www.wias-berlin.de/people/koenig/www/GA.pdf Korollar 1.3.2

I thought about doing a Laplace Transformation, but I sucked. Furthermore, I tried to change dx to a Dirac- measure to connect it to the Varadhan's Lemma (the rate function is 0). I would like to use the senctence stated above (Lemma 1.3.1), but it didn't work. How can I write an integral as a sum?

Kind regards, mimi

Answer 1

You can approximate the integral by Riemann sums:

$$\int_0^1e^{nf(x)}\,dx\approx \frac{1}{n}\sum_{k=1}^ne^{nf(x_k)}$$,

where it is assumed that the interval $[0,1]$ is partitioned into $n$ intervals of length $1/n$ each, and $x_k$ are points chosen from the partition-intervals. Then you can use the lemma, together with an additional argument, to conclude the result given as korollar 1.3.2.