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Lebesgue integral question concerning orders of limit and integration
I happened upon the following integration exercise. I don't know how standard it is, but it caught my eye, and I am wondering how best to approach it.
Suppose that $\phi \in L_1(\mu)$ is nonnegative, with $\alpha \in \mathbb{R}$ such that $\alpha>0$. Compute the value of the following:
$\lim_{n \to \infty} n \cdot \int \log (1 + (\phi(x)/n)^\alpha) d\mu (x).$
The author gives a hint: separately treat the cases $\alpha < 1$, $\alpha = 1$, $\alpha > 1$. I am wondering if such case-work will demonstrate divergence in at least one case. That aside, this is the first time I came across an Lebesgue integration exercise with a logarithm present, and it is throwing me! If anyone visiting the site today has performed a computation at this level, and would be up for an assist, I could use the help and would appreciate it! Thanks.