Given the space $(\mathbb{R},\mathbb{B},\mu)$ where $\mathbb{B}$ is the $\sigma$-algebra on $X$ and $\delta$ is the dirac-measure (1 if $a\in B\in\mathbb{B}$ and zero otherwise), I cannot understand the difference in the following two arguments.
The completion would be: $$\mathbb{B}_{\delta_{a}}=\{B\cup N\;|\;B\in\mathbb{B},N\in\mathbb{N}_{\delta_{a}}\}\subseteq\{B\cup \{a\}\;|\;B\in\mathbb{B},a\notin B\}=\mathcal{P}(X)$$ however, why is it wrong to say: $\mathbb{B}(\mathbb{R})\cup \mathcal{P}(\mathbb{R}$ \ $\{a\}\}=\mathcal{P}(X),$ as $\mathbb{N}_{\delta_{a}}=\{N\subseteq \mathbb{R}|N\subseteq \mathbb{R}$ \ $\{a\}\}?$
If $\mathcal A $ and $\mathcal B $ are two families of sets there is a difference between $\mathcal A \cup \mathcal B $ and $\{A\cup B: A\in \mathcal A, B \in \mathcal B \}$. So you cannot write the completion as $\mathcal B (\mathbb R) \cup \mathcal P (\mathbb R\setminus \{a\})$.