I have the following problem from Folland:
Let $X = [0, 1]$, $\mathcal{M} = \mathcal{B}_{[0, 1]}$, $m = $ Lebesgue measure and $\mu = $ counting measure.
- $m \ll \mu$ but $dm \neq f \, d\mu$ for any $f$.
- $\mu$ has no Lebesgue decomposition with respect to $m$.
I think I might be understanding counting measure incorrectly, because it seems to me that $m \ll \mu$ is not true, because any Borel subset of the open interval $(0, 1)$ would have counting measure zero because it does not contain any integers, but clearly could have positive Lebesgue measure.
Best Answer
The counting measure does exactly what it says: it counts the number of elements in a set. So any infinite set has counting measure $\infty$, while the measure of any finite set is its cardinality. It has nothing to do with integers in particular.
It follows that any measure is absolutely continuous with respect to counting measure, while very few functions are integrable.