I have two similar questions:

1) Let $X$ and $Y$ be two measure spaces. Suppose for every point $x \in X $ there exists a set $ \mathcal{U}_x \subset Y $ of full measure in $Y$. Suppose $V \subset X $ has full measure in $X$. Consider the subset of $X \times Y$ given by $$ W := \cup_{x \in V} \{x\} \times \mathcal{U}_x \subset X \times Y $$ Does $W$ have full measure?

2) Same question as above, but $X$ and $Y$ are complex algebraic varieties and we say that a set has full measure if the complement is contained in a finite union of subvarieties of strictly smaller dimension.

The motivation for the question is as follows: How do you rigorously prove this fact: Three generic lines in $\mathbb{C}^2$ do not intersect. The way I want to prove it is using these three facts:

a) Two generic lines in $\mathbb{C}^2$ intersect in only one point.

b) Given a specific point a generic line does not pass through that point.

Can I conclude that three generic lines do not intersect generically simply using a) and b)? The idea being that let $A$ be the space of lines in $\mathbb{C}^2$. Let $$ X := A \times A, \qquad Y:= A, \qquad V:= A \times A-\Delta_{A} $$ where $\Delta_A$ is the diagonal of $X$. Now given $(L_1, L_2) \in V $ we define $\mathcal{U}_{L_1,L_2} \in Y $ to be the space of lines that do not pass through the intersection of $L_1$ and $L_2$. The set $W$ as defined earlier is precisely the space of three lines which do not intersect. Why does this set have full measure?

More precisely, what is the easiest way to see this space has full measure, preferably just using the fact that $V$ and $\mathcal{U}_{L_1, L2} $ have full measure.