Let $X\subset R$ and $Y \subset R$ where X and Y have finite cardinalities. Let also, $a,b \in R$. How to show that $|aX+bY|=|X||Y|$ almost everywhere (measure of $(a,b) \subset R^2$ such that $|aX+bY|<|X||Y|$ is 0) ? Where we define $|aX+bY|=\{ax+by:x \in X, y \in Y \}$.
More, specifically let $X=\{x=d_x*z: z \in \mathcal{Z}, d_x \in \mathcal{R}\}$ (so, it's a lattice. Where each point is $d_x$ away from one another.) The above is easy if $X=Y$ then the proof of $|aX+bX|=|X||X|$ almost everywhere is fairly simple. Now what if $Y$ is lattice with $d_y \neq d_x$. How to show it then? Thanks in advance.