I am trying to prove that the following probability measure given in my book (Rohtagi) is countably additive. My analysis is a bit rusty so if someone can explain to me the properties of integrals involved in proving that the probability measure is indeed countably additive I would be very grateful.
Let $(\Omega=(0,\infty),\mathbb{B})$ be a sample space. Here $\mathbb{B}$ is the Borel $\sigma$-field on $\Omega$. (The little bit of measure theory I remember tells me that a $\sigma$-field is a non-empty collection of subsets of $\Omega$ which is closed under countable union, complements and contains $\emptyset$. A Borel $\sigma$-field on $\Omega$ is the smallest $\sigma$-field on $\Omega$ containing all intervals). Let $P$ be defined for each interval $I$ as $PI = \int_I e^{-x}dx$. Now I want to prove that $P$ is countably additive. I am not sure how to do that for an arbitrary disjoint sequence of Borel sets.