- #1

fabiancillo

- 27

- 1

Let $(X,\mathcal{B} , \mu)$ a measurement space, consider

$\bar{\mathcal{B}} = \{ A \subseteq{X} \; : \; A\cap{B} \in \mathcal{B}$ for all that satisfies $\mu(B) < \infty \}$, and

for $A \in \bar{\mathcal{B}}$ define

$\bar{\mu}(A) = \left \{ \begin{matrix} \mu (A) & \mbox{if }A \in \mathcal{B}

\\ +\infty & \mbox{if }A \not\in\mathcal{B}\end{matrix}\right. $

Prove that $(X,\bar{\mathcal{B}} ,\bar{\mu})$ X is a measure space and $\mathcal{B} \subseteq{} \bar{\mathcal{B}} $