Vanishing measure of a set with codimenon 2

[cduston]

Hey everyone,
I am integrating something (specifically 2-forms, but I think this is a general statement) over a set B of (real) codimension 2 in a 4-manifold (CP_2). I've been told that the measure of a set of codimension 2 will vanish, but I don't really understand why. I've been thinking about exterior products and hodge duals but I can't seem to understand it from that direction. Does anyone have any insights?

Thanks!

 

[n_bourbaki]

This isn't a statement about differential manifolds or anything like it - it's just measure theory.

I'm not going to attempt a rigorous proof, but explain by example/analogy. Think of a line in R^3, which has codimension 2. The lebesgue measure of that is zero. This you can prove rigorously, but non-rigorously something that is locally 1-dimensional has no (3-dimensional) volume.

 

[cduston]

Ok I see that's actually pretty simple. I guess I was thinking that integrating something like a line in R^3 should be dl (like a 1D integral), which isn't generally zero but if you took the volume integral dV=dxdydz (or whatever) over the line you would get zero. Ok, thanks!