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Could give some examples of nonnegative measurable function $f:\mathbb{R}\to[0,\infty)$, such that its integral over any bounded interval is infinite?

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    Would a simple counting function do?2011-03-01

2 Answers 2

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The easiest example I know is constructed as follows. Let $q_{n}$ be an enumeration of the rational numbers in $[0,1]$. Consider $g(x) = \sum_{n=1}^{\infty} 2^{-n} \frac{1}{|x-q_{n}|^{1/2}}.$ Since each function $\dfrac{1}{|x-q_{n}|^{1/2}}$ is integrable on $[0,1]$, so is $g(x)$ [verify this!]. Therefore $g(x) < \infty$ almost everywhere, so we can simply set $g(x) = 0$ in the points where the sum is infinite.

On the other hand, $f = g^{2}$ has infinite integral over each interval in $[0,1]$. Indeed, if $0 \leq a \lt b \leq 1$ then $(a,b)$ contains a number $q_{n}$, so $\int_{a}^{b} f(x)\,dx \geq \int_{a}^{b} \frac{1}{|x-q_{n}|}\,dx = \infty.$ Now in order to get the function $f$ defined at every point of $\mathbb{R}$, simply define $f(n + x) = f(x)$ for $0 \leq x \lt 1$.

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    @kahen: I agree.2011-03-01
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See exercise 26 (c) on p. 327 here.

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    See kahen's comment below Theo's answer...2011-08-02