Completion of Borel Algebra with point mass measure at 0. Trying to find the complition of $(\mathbb{R},\cal{B}(\mathbb{R}),\mu_0)$ where $\cal{B}(\mathbb{R})$ is Borel Algebra and $\mu_0(E)=1$ if $0\in E$ and zero otherwise.
So I know I need to include the subsets of zero measured sets from the algebra to form a new algebra. But Borel algebral confuses me a lot since it can be generated by open or closed or $[a,b)$ or $(-\infty,b)$ type of sets.
I think the answer is $\cal{B}(\mathbb{R}) \cup \cal{P}(\mathbb{R}\backslash \lbrace 0 \rbrace)$.
Am I right?

The completion should be a $\sigma$-algebra (call it $\Sigma$) containing $\mathcal{P}(\mathbb{R} \setminus \{ 0 \}) \cup \mathcal{B}(\mathbb{R})$. But now, one can easily see that this $\sigma$-algebra must be the whole $\mathcal{P}(\mathbb{R})$. In fact, for all $S \in \mathcal{P}(\mathbb{R})$:


*

*If $0 \notin S$, then $S \in \mathcal{P}(\mathbb{R} \setminus \{ 0 \} ) \subseteq \Sigma$

*If $0 \in S$, then $S =  (S \setminus \{ 0 \}) \cup \{ 0 \} \in \Sigma$ because $(S \setminus \{ 0 \}) \in \mathcal{P}(\mathbb{R} \setminus \{ 0 \} ) \subseteq \Sigma$, and $\{ 0 \} \in \mathcal{B}(\mathbb{R}) \subseteq \Sigma$


Any way, you get $\mathcal{P}(\mathbb{R}) \subseteq \Sigma$
