Outer Measure Restricted to Borel Sets

• categories: Measure Theory, Theorem

Outer Measure Restricted to Borel Sets

If an Outer Measure ${\mu }^{\ast }$ is restricted to the Borel sigma algebra $\mathcal{B}$, it becomes a proper Measure. That is, the function $\mu :\mathcal{B}\to [0,\infty ]$ defined by $\mu (A)={\mu }^{\ast }(A)\text{for all}A\in \mathcal{B}$ is a measure.