# If $R$ be a Union of zero measure sets , what is the cardinal of index set? [duplicate]

If $R$ be a Union of zero measure (lebesgue) sets , what can we say about the cardinal of index set? Does this question related to continuum hypothesis? Thanks.

## marked as duplicate by BrianO, Claude Leibovici, user91500, user228113, Eric WofseyFeb 4 '16 at 7:26

• What have you done so far to solve it? Where are you stuck? – Nate 8 Feb 4 '16 at 3:34

Suppose $\kappa$ is a cardinal such that $\mathbb{R} = \bigcup_{\alpha < \kappa} A_\alpha$, where each $A_\alpha$ is measure zero. Then $\kappa$ is greater than equal to a cardinal called $\text{cov(null)}$, which is by definition the smallest cardinal $\kappa$ such that there exists a family $\{A_\alpha : \alpha < \kappa\}$ of null sets whose union is $\mathbb{R}$.
It is clear that $\text{cov(null)} > \aleph_0$. Hence if the continuum hypothesis is true then, $\text{cov(null)} = 2^{\aleph_0}$.
However, there are models (where the continuum hypothesis is false) in which $\text{cov(null)} < 2^{\aleph_0}$.
• @hctb By the way, one explicit example is to start off the with the constructible universe $L$ as forcing using the $\omega_2$ length countable support iteration of Sacks forcing (perfect trees). – William Feb 4 '16 at 4:25
• @hctb Also it is possible for the continuum hypothesis to be false and $\text{cov(null)} = 2^{\aleph_0}$. For this you can use an $\omega_2$ length countable support iteration of Random forcing over $L$. – William Feb 4 '16 at 4:26