(The following question arose in a joint research with Adam Przeździecki and Boaz Tsaban.)

For a $\sigma$-ideal $\mathcal{I}$ of subsets of the unit interval $[0,1]$, define
$$\newcommand{\card}[1]{\left|#1\right|}\newcommand{\cov}{\operatorname{cov}}\newcommand{\sub}{\subset} \cov(\mathcal{I}):=\min\{\card{\mathcal{A}}: \mathcal{A}\sub \mathcal{I}, \bigcup\mathcal{A}=[0,1]\}. $$ Let $\mathcal{E}$ be the family of all $F_\sigma$ Lebesgue null subsets of the unit interval, and $\mathcal{N}$ be the family of all Lebesgue null subsets of the unit interval.

1. Let $\kappa_\mathcal{E}$ be the minimal cardinal number such that some measure one set is covered by $\kappa_\mathcal{E}$ elements of $\mathcal{E}$.
2. Similarly, let $\kappa_\mathcal{N}$ be the minimal cardinal number such that some measure one set is covered by $\kappa_\mathcal{N}$ elements of $\mathcal{N}$.

We have $\kappa_\mathcal{E}\leq \cov(\mathcal{E})$, and $\kappa_\mathcal{N}=\cov(\mathcal{N})$.

Question: Is it provable that $\kappa_\mathcal{E}=\cov(\mathcal{E})$?

(The following question arose in a joint research with Adam Przeździecki and Boaz Tsaban.)

For a $\sigma$-ideal $\mathcal{I}$ of subsets of the unit interval $[0,1]$, define
$$\newcommand{\card}[1]{\left|#1\right|}\newcommand{\cov}{\operatorname{cov}}\newcommand{\sub}{\subset} \cov(\mathcal{I}):=\min\{\card{\mathcal{A}}: \mathcal{A}\sub \mathcal{I}, \bigcup\mathcal{A}=[0,1]\}. $$ Let $\mathcal{E}$ be the family of all $F_\sigma$ Lebesgue null subsets of the unit interval, and $\mathcal{N}$ be the family of all Lebesgue null subsets of the unit interval.

1. Let $\kappa_\mathcal{E}$ be the minimal cardinal number such that some measure one set is covered by $\kappa_\mathcal{E}$ elements of $\mathcal{E}$.
2. Similarly, let $\kappa_\mathcal{N}$ be the minimal cardinal number such that some measure one set is covered by $\kappa_\mathcal{N}$ elements of $\mathcal{N}$.

We have $\kappa_\mathcal{E}\leq \cov(\mathcal{E})$, and $\kappa_\mathcal{N}=\cov(\mathcal{N})$.

Question 1: Is it provable that $\kappa_\mathcal{E}=\cov(\mathcal{E})$?

Question 2: Does the cardinal number $\cov(\mathcal{E})$ change if we work in a closed positive subset of the unit interval instead of the whole interval?

A negative answer for Question 2 implies a positive answer for Question 1.