Let $X$ be a measure space, and $\{f_n\}$ a decreasing sequence of nonnegative measurable functions $X\to\mathbb{R}$. Is it possible that none of $f_n$ is integrable (i.e., $\int f_n=\infty$ for all $n$), yet $\inf f_n$ is integrable (i.e., $\int \inf f_n<\infty$) ?
This question originates from the following observation:
Let $\{f_n\}$ be an increasing sequence of nonnegative measurable functions $X\to\mathbb{R}$. If none of $f_n$ is integrable (i.e., $\int f_n=\infty$ for all $n$), then clearly $f$ is not integrable (i.e., $\int f=\infty$).
I wondered if it still holds for decreasing sequences. When I tried to prove it, the same method of proof somehow didn't seem to work; and now I'm beginning to suspect that it may be false.
This is just an elaboration on @geetha290krm's answer: Let $\{f_n\}_{n=1}^\infty$ be a sequence of functions defined by $$f_n:(0,1]\to\mathbb{R},\qquad x\mapsto\frac{1}{nx}.$$ Then none of $f_n$ is integrable (with respect to the Lebesgue measure), but $\inf f_n=0$ is integrable.