Suppose $\kappa = \mathfrak{c} =2^{\aleph_0}$ is a real valued measurable cardinal with a witnessing measure $m:\mathcal{P}(\kappa) \to [0, 1]$ - So $m$ is a diffused (points have zero measure) $\kappa$-additive measure on $\mathcal{P}(\kappa)$. Let $\mu = \otimes_{n < \omega} m$ be the countble product measure - So $\mu$ is defined on the sigma algebra generated by the family $\{\Pi_{n < \omega} X_n : (\forall n)(X_n \subseteq \kappa)\}$. Let $\mathcal{N}$ be the sigma ideal of $\mu$-null subsets of $\kappa^{\omega}$.

What can be say about the cardinal invariants associated with the ideal $\mathcal{N}$? For example, can either one of $add(\mathcal{N}), non(\mathcal{N}), cov(\mathcal{N})$ be $\aleph_1$?

Definitions: $non(\mathcal{N})$ is the least size of a set not in $\mathcal{N}$, $add(\mathcal{N})$ is the least size of a family $F \subseteq \mathcal{N}$ whose union is not in $\mathcal{N}$ and $cov(\mathcal{N})$ is the least size of a family of members of $\mathcal{N}$ whose union is $\kappa^{\omega}$.