I have been given the following question:
Let $\mu$ be a Borel measure on $[1, \infty)$ given by the density function 1/x with respect to the 1-dimensional Lebesgue measure. Is measure $\mu$ finite or $\sigma$-finite?
Since the outputs of the density function contains numbers within the rational numbers, which are countable, and furthermore of finite measure, I believe it fits the description of $\sigma$-finite, but I'm not sure?
Furthermore, I must consider a function $f : [1, \infty)→ \mathbb R$ such that $f(x)= e^{-x}cos(x)$. I need to find out whether or not the integral $∫_{[1, \infty)}fdμ$ exists, I don't have to evaluate it, but just give precise arguments.
I'm pretty sure the integral exists and is equal to 1/2, but I don't know how to give a more general proof, of how to show whether or not it exists :-)