Looking for example $\lim_{\alpha \to 1^+} \int_A f^\alpha \,d\lambda \ne \int_A f \,d\lambda$ where $\mu(A) < \infty$

Question

I'm looking for a summable non-negative function $f: \Bbb{R} \to [0,\infty)$ and a measurable set $A$ with finite measure such that

$$\lim_{\alpha \to 1^+} \int_A f^\alpha \,d\lambda \ne \int_A f \,d\lambda$$

I believe that $f^\alpha$ should not be summable for every $\alpha>1$ but I couldn't find such an example.

Answer 1

Consider $A=(0,1]$ and $$f(x) := \frac{1}{x} \frac{1}{(\log x)^2}.$$ Then $\int_A f^{\alpha} d\lambda = \infty$ for all $\alpha>1$, but $$\int_A f \, d\lambda <\infty.$$