show that $f_k$ is lebesgue integratable

Question

Let $f : \mathbb{R}^n \to [0;\infty[$ be a non-negative, lebesgue integratable function and for every $k \in \mathbb{N}$ let the functions $f : \mathbb{R} \to \mathbb{R}$ be $f_k(x) = k\cdot \log(1+\frac{f(x)}{k})$. Show that f is lebesgue integratable for every k and that $lim_{k \to \infty} \int f_k dv_n = \int f dv_n$.

So, my idea was to show that the set $A_c := \{x \in \mathbb{R}: f_k(x) \geq c_k := k \cdot \log(1)\}$ is measurable and then concluding f is measurable. But it seems hard to show that $A_c$ is either closed or open. I haven't thought about the second part yet, where one needs to show that the $\lim$ of $f_k$ for $k \to \infty$ is equal to $f$. Help there would be appreciated too since I have no idea how to solve that either.

Answer 1

$k \log (1+\frac a k) \to a$ for every real number $a$.

From the inequality $\log (1+x) \leq x$ for $x \geq 0$ we see that $0\leq f_k(x) \leq f(x)$. Also $f_k (x) \to f(x)$ for all $x$. Hence DCT can be applied.

Answer 2

HINT: for any $c\geqslant 0$ the sequence $(c_n)$ defined by $c_n:=\left(1+\frac{c}{n}\right)^n$ is increasing and converge to $e^c$.