/sci/ - Science & Math

>>11890000
Set [math]f_k (t) : [0, 1] \rightarrow \mathbb{R}[/math] by [math]f_k(t) = f(k+t)[/math], for [math]k \in \mathbb{N}[/math].
Call the usual measure on [math]\mathbb{N}[/math] by [math]\lambda[/math].
Then [math]\int_{ \mathbb{R}} f ~ d \mu = \int _{[0, 1] \times \mathbb{N}} f_k ~ d ( \mu \times \lambda) = \int_{[0, 1]} \int _{ \mathbb{N}} f_k ~ d \lambda d \mu = \infty[/math], if [math]f \geq 0[/math], by Fubini.
I could be misusing Fubini, tho.