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 $\mathbb{R}^{m\times n}$
* $\partial A\_j = A\_j\cap B$.



What does it mean to say "almost always" ?
I see some contradiction in your hypotheses: since the $A\_j$'s are disjoint and have non empty interiors, the union should be at most countable (there can't exist more $A\_j$'s than the cardinal of the set of rational points in $\mathbb{R}^{m\times n}$).


