Generalized Cantor set

Choose a strictly decreasing positive sequence $q_0=1,q_1,q_2,\dots$. Begin with $C_0=[0,1]$ and at step $n+1$ keep two closed end intervals of length $q_{n+1}/2^{n+1}$ from every component interval of $C_n$. This constructs a compact set $C={\bigcap }_n{C}_n$ withThe second equality is continuity from above of a measure. Every stage has positive gaps and maximum component length at most $2^{-n}$, so $C$ contains no nondegenerate interval and is a nowhere dense set. It has no isolated points: both endpoints of the stage-$n$ component containing a point remain in $C$, and at least one distinct endpoint is within $2^{-n}$. Thus its topological sparseness is compatible with positive Lebesgue measure.

Taking $q_n=(2/3{)}^n$ gives the usual Cantor set of measure zero. Taking $q_n=1/2+2^{-(n+1)}$ gives a generalized construction of measure $1/2$. This is one useful family of generalized constructions; the removed intervals need not have the same proportions at every stage.