Note: In the following, all submanifolds are assumed to be smoothly embedded, i.e. the image of a smooth embedding.
Definition: A Cantor subset of a topological space $X$ is a closed, nowhere dense subset of $X$ with no isolated points.
Problem set up:
Let $M$ be a compact, finite dimensional smooth manifold without boundary, and let $K$ and $N$ be submanifolds of $M$ with $\dim K + \dim N > \dim M$.
Suppose that $K \cap N$ is a Cantor subset of another submanifold $J$ of $M$ with $1 \leq \dim J \leq \dim K + \dim N - \dim M$.
Denote by $T$ the subset of $K \cap N$ on which $K$ and $N$ intersect transversally.
Question: Can $T$ have nonzero measure in $J$?