Hello guys,
I would like to ask following:

Special case of Vitali theorem says, that if A\subset\mathbb{R} is set, for which there holds \lambda (A)<\infty
(where \lambda means Lebesgue measure),
and system of closed intervals J=\{J_\alpha\}_{\alpha\in\mathcal{A}} covers A in Vitali sense (i.e. \forall \delta > 0 \,\,\forall x\in A\,\,\exists I\in J\,:\,x\in I \wedge \lambda (I)<\delta ),

a) For any given \varepsilon > 0 there exists finite amount of pairwise disjoint intervals I_1,I_2,\dots,I_N\in J, such that

\lambda^{*}\Big(A-\bigcup_{i=1}^N I_i\Big)<\varepsilon

b) There exists countable substystem  J'=\{I_n\}_{n\in\mathbb{N}} of J (intervals in J' are pairwise disjoint), such that

\lambda^{*}\Big(A-\bigcup_{n\in\mathbb{N}} I_i\Big)=0

where \lambda^* means outer Lebesgue measure.

I want to ask if there's any easy way how to prove, that
a) => b) and b)=> a), or the proofs have to be done separately.

Thank you very much