@skewray (2/2)
An example more relevant to your proposed extra rule is the following. Take a 3D spherical octant, with radius r, latitude φ, longitude λ as coordinates:
r ∈ [0, 1], φ ∈ [0, π/2], λ ∈ [0, π/2]
and the probability density
p(r, φ, λ) dr dφ dλ = 2 cos(φ)/π dr dφ dλ .
Let's take the 1D set B defined by {φ = π/2} (line from centre to north pole), and the 2D set A defined by {λ = 0}. These sets satisfy your requirement B ⊆ A. It is easy to verify that both have zero measure.
What's the value of p(A|B)? A priori undetermined. Let's calculate it through some limit. For instance, we could define B as the limit of the region π/2 - ε < φ ≤ π/2, and A as the limit of the region 0 ≤ λ < δ. We find
p[A(δ) | B(ε)] = p[A(δ) ∩ B(ε)] / p[B(ε)] =
p(0 ≤ r ≤ 1, π/2 - ε < φ ≤ π/2, 0 ≤ λ < δ) / p(0 ≤ r ≤ 1, π/2 - ε < φ ≤ π/2, 0 ≤ λ < π/2) =
2 δ/π
and in the limit δ→0 we get p(A|B) = 0.
You may object that this particular choice of limit way of defining B and A is arbitrary; and you're completely right. We could use other limiting ways, and you can check that you could get other limit values for p(A|B), also different from 0 or 1.
This is exactly the point: what's "arbitrary" or not depends on the specific context. In some applications the limits above can make sense, and therefore p(A|B) = 0 makes sense there; in other applications, they may be unnatural. This is why I like that p(A|B) (with B ⊆ A, both zero measure) is a priori undefined; it means "please provide more information or a context, to get a definite answer".
Apologies if my discussion sounds patronizing, that's really not my purpose! The reason why I got engaged with your initial post is that I was also, for quite some time, completely unsatisfied with the indeterminacy of conditional probability in situations like these. I didn't get why it should be so, since in some cases some definitions seemed natural to me. So I get where you're coming from! :) One funny thing is that in the case you're considering, my intuition told me "p(A|B) = 0" was "right" instead. It's when I started to do concrete applications that I realized things can go in very different directions, for very natural reasons in every specific application (shock waves, shells in continua, condensation points, sets in manifolds with generic Riemannian metrics...), that's when I learned to appreciate the indeterminacy, as it was warning me "get more info first".
#probability #mathematics