THE σ-ALGEBRA GENERATED BY THE JORDAN SETS IN ℝn

Abstract

Let $I$ be a non-degenerate interval in ${ℝ}^n$; $J$, $B$ and $L$ are the Jordan, Borel and Lebesgue sets respectively in $I$; $\sigma (J)$ is the $\sigma$-algebra generated by $J$; $F_0=\{n\hspace{0.17em}:\hspace{0.17em}n\text{ is a subset of an }{F}_{\sigma }\text{ subset of }I\text{ having Lebesgue measure zero}\}$; $S_0=\{b\cup n\hspace{0.17em}:\hspace{0.17em}b\in B,n\in F_0\}$; $S_1=\{b\cup n:b\in B,n\text{ is a first category subset of }I\text{ having Lebesgue measure zero}}$. The symbol “$\subset$” denotes “proper subset”.

THEOREM 1. $\sigma (J)=S_0$.

THEOREM 2. $B\subset \sigma (J)\subset S_1$.