In reply to Gerwin.

2. Measure theory is the mathematical foundation of probability. "Measure zero" means that lines and points have zero area. It doesn't matter whether you include lines and points when computing areas. For example, the area of the unit square and its boundary is 1. But also, the area of the unit square WITHOUT its boundary is 1! This shows that Area([0,1]x[0,1]) = Area([0,1]x[0,1]) = 1.

3. No, there is no difference. The probability of randomly selecting a point that is on the boundary is equal to the measure (area) of the boundary, which is zero.