applying convergence theorem to a integral with the log-function

Question

Let $f:\mathbb{R}\to\mathbb{R}_{\ge 0}$ be a non-netagtive function, $f\in\mathcal{L}^1(\mathbb{R})$ (i.e. Lebesgue integrable).

What is $$\lim\limits_{n\to\infty} n\int_{\mathbb{R}} \ln\Big(1+\frac{f}{n}\Big) ds ?$$

It is $n\ln(1+\frac{f}{n})=\ln\Big((1+\frac{f}{n})^n \Big )$ which converges pointwise to $\ln(e^{f})=f$, as $n\to\infty$.

is there a way to apply the monotone convergence theorem (Beppo Levi) to calculate the limit?

Answer 1

If $t>0$ then $$\ln(1+t)\le t.$$Hence $$\left|n\ln(1+f/n)\right| \le n\ln(1+|f|/n)\le |f|.$$ So another one of the big theorems can be applied here...