Extension from Pi-System

Definition

Probability Measure

If $\mathcal{A}$ is a Pi-System on the Probability Space $(\Omega ,\sigma (\mathcal{A}),P)$, then the Probability Measure $P:\sigma (\mathcal{A})\to [0,1]$ is uniquely determined by the probability measure-like function on the Pi-System $P^{\ast }:\mathcal{A}\to [0,1]$

Finite Measure

If $\mathcal{A}$ is a Pi-System containing the universal set $\Omega$ on the finite Measure Space $(\Omega ,\sigma (\mathcal{A}),\mu )$, then the Finite Measure $\mu :\sigma (\mathcal{A})\to [0,\mu (\Omega )<\infty ]$ is uniquely determined by the finite measure-like function on the Pi-System ${\mu }^{\ast }:\mathcal{A}\to [0,\mu (\Omega )<\infty ]$

Sigma-Finite Measure

If $\mathcal{A}$ is a Pi-System on the Sigma-Finite Measure Space $(\Omega ,\sigma (\mathcal{A}),\mu )$, then the Sigma-Finite Measure $\mu :\sigma (\mathcal{A})\to [0,\infty ]$ is uniquely determined by the sigma-finite measure-like function on the Pi-System ${\mu }^{\ast }:\mathcal{A}\to [0,\infty ]$

Facts

If the two measures ${\mu }_1,{\mu }_2$ defined on a Sigma-Field Generated by A Families of Sets $\sigma (\mathcal{A})$ satisfy the following conditions, then they are the same ${\mu }_1={\mu }_2$.

• $\forall A\in \mathcal{A}:{\mu }_1(A)={\mu }_2(A)$
• ${\mu }_1(\Omega )={\mu }_2(\Omega )<\infty$