[@Kolmogorov1956] laid the foundation of axiomatic probability theory.
Sample space $\Omega$ is a set of elementary events. A sigma-algebra $\Sigma$ of a sample space consists of all the events that can be measured using some instrument.
Probability measure $P$ of a measurable sample space $(\Omega, \Sigma)$ is a normalized finite measure: $P(\Omega) = 1$. Any finite measure can be normalized into a probablity measure.
Probability space $(\Omega, \Sigma, P)$ is a sample space $\Omega$ assigned with a probability measure $P$ on its measureable sets $\Sigma$.