Hello guys,

I would like to ask following:

Special case of Vitali theorem says, that if $\displaystyle A\subset\mathbb{R}$ is set, for which there holds $\displaystyle \lambda (A)<\infty$

(where $\displaystyle \lambda$ means Lebesgue measure),

and system of closed intervals $\displaystyle J=\{J_\alpha\}_{\alpha\in\mathcal{A}}$ covers $\displaystyle A$ in Vitali sense (i.e. $\displaystyle \forall \delta > 0 \,\,\forall x\in A\,\,\exists I\in J\,:\,x\in I \wedge \lambda (I)<\delta$ ),

then:

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

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

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

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

where $\displaystyle \lambda^*$ 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