I have a set, $A$, of $m \times n$ matrices with certain properties and a subset $B$ of $A$. I would like to say that when randomly selecting such a matrix, I am "almost always" never in $B$. I can show that $A$ is the disjoint union of subsets, $A_j$, such that for each $A_j$, the measure of $A_j\cap B$ is 0. However, this union is uncountable. Is this enough to reasonably conclude that my randomly selected matrix will "almost always" never be in $B$?

Thanks for the help!

Best, Julie

Additional info: Each $A_j$ has the following properties :

- the interior of $A_j$ is a nonempty open subset of $\mathbb{R}^{m\times n}$
- $\partial A_j = A_j\cap B$.