# Counter-example to homogeneity of unsigned lower Lebesgue integral.

I know this is sort of a nonissue, but in one of the exercises the author asks us to prove

If $0\leq c<+\infty$, then $\int cf=c\int f$ where $f:\mathbb{R}^{d}\to[0,+\infty]$ (note $f$ is not assumed measurable, just unsigned).

The integral is the "lower unsigned Lebesgue integral" as is usually defined, except that as stated $f$ is any unsigned function.

This is easy enough to show, however, he adds a parenthetical note stating: "(The case $c=+\infty$ unfortunately is not true, but this is somewhat tricky to show.)

I can't think of a counter-example when the convention $\infty\cdot0=0$ is used. Anyone have an idea?

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