Let $\mu$ be a strictly positive measure, and let $f:X\rightarrow[0,\infty]$ be a an function such that $\int_Xfd\mu=1$. Calculate the following limit:

$\lim_{n\rightarrow \infty}n\int_X\log\biggl(1+\frac{f(x)}{n}\biggl)d\mu$


Note that $\left|n\log\left(1+\frac{f(x)}n\right)\right|\le|f(x)|$, so we can use $|f(x)|$ as a dominating function for Dominated Convergence.

Pointwise, $$ \lim_{n\to\infty}n\log\left(1+\frac{f(x)}n\right)=f(x) $$ Therefore, by Dominated Convergence, we have $$ \lim_{n\to\infty}n\int_X\log\left(1+\frac{f(x)}n\right)\,\mathrm{d}\mu =\int_Xf(x)\,\mathrm{d}\mu=1 $$


My approach:

Since $\log(x+1)\leq x$ for all $x \in [0,\infty]$ and $f \leq 0$ then for every $n>0$: $n\log\biggl(1+\frac{f(x)}{n}\biggl) \geq 0$ and therefore: $\biggl |n\log\biggl(1+\frac{f(x)}{n}\biggl)\biggl| \leq n \cdot \frac{f(x)}{n}=f(x)$.

Morevoer, because $f$ is non-nogative, then $\int_X|f|d\mu=\int_Xfd\mu=1<\infty$.

$\implies f \in L^1(X)$. i.e, integrable over X.

Clearly, from the monotony of the integral:

$\int_X\biggl|n\log\biggl(1+\frac{f(x)}{n}\biggl)\biggl|d\mu \leq \int_Xf(x)d\mu=1<\infty.$ So, $n\log\biggl(1+\frac{f(x)}{n}\biggl) \in L^1(X)$ as well.

Now, we notice that: $\lim_{n\rightarrow\infty}n\log\biggl(1+\frac{f(x)}{n}\biggl)=\lim_{n\rightarrow\infty}\log\biggl(\biggl(1+\frac{f(x)}{n}\biggl)^n\biggl) = \log\biggl(\lim_{n\rightarrow\infty}\biggl(1+\frac{f(x)}{n}\biggl)^n\biggl)$

$= \log\biggl(\exp(f(x))\biggl)=f(x)$

$\implies$ By the dominant convergence theorem, we get:

$\lim_{n\rightarrow \infty}n\int_X\log\biggl(1+\frac{f(x)}{n}\biggl)d\mu = \lim_{n\rightarrow \infty}\int_Xn\log\biggl(1+\frac{f(x)}{n}\biggl)d\mu$

$= \int_X\lim_{n\rightarrow \infty}n\log\biggl(1+\frac{f(x)}{n}\biggl)d\mu = \int_Xf(x)d\mu = 1$

Is my solution is fine? Did I miss anything that is worth explaining?

I would love for some feedback!

Thank you very much :)


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