A problem in measure theory (outer measure) Show that for every set $A = P(\mathbb{R})$ (power set) there exists $B \in \mathbb{B}(\mathbb{R})$ (Borel set) s.t $A \subset B$ $\lambda^{*}(A) = \lambda (B)$ and $\lambda (N) = 0 \ \forall N \in \mathbb{B}(\mathbb{R})$ with $N = \mathbb{B} \backslash A$
where 
$$
\lambda^{*}(A) = \inf \{ \sum_{j \in \mathbb{N}} \lambda (I_j) | I_j \in J, j \in \mathbb{N}, A \subset \cup_{j \in \mathbb{N}} I_j \} 
$$
where $J$ is the half open rectangles in $\mathbb{R}$ i.e $[a,b)$
If I start looking at the case when $\lambda^{*} (A) < \infty$. By definition of $\inf$ $\exists \epsilon >0$ s.t for $S \subset \cup_{j \in \mathbb{N}} I_j$ then $\lambda (S) - \lambda^{*} (A) < \epsilon$. How can I proceed?

Hint.
For $n \in \mathbb N$ there exists $\displaystyle B_n =\bigcup_{j \in \mathbb N} I_{j,n}$ with $I_{j,n} \in J$ such that
$$\lambda(B_n)-\lambda^*(A)\le \frac{1}{n}$$ Then define
$$\displaystyle B=\bigcap_{n \in \mathbb N} B_n$$
