A multivariate generalization of a theorem of R. H. Farrell

Abstract:

Let $(\Omega ,\mathfrak {S},\mu )$ denote a finite measure space and ${T_j}:\Omega \to [{a_j},{b_j}](\mathfrak {S},\mathfrak {B} \cap [{a_j},{b_j}])$-measurable functions, $j = 1, \ldots ,n$. Then the algebra of functions generated by $1,{T_1}, \ldots ,{T_n}$ is a dense subset of ${\mathcal {L}_1}(\Omega ,\mathfrak {S},P)$ if and only if for any $A \in \mathfrak {S}$ there exists some $B \in {({T_1}, \ldots ,{T_n})^{ - 1}}({\mathfrak {B}^n} \cap [{a_1},{b_1}] \times \cdots \times [{a_n},{b_n}])$, such that $\mu (A\Delta B) = 0$ is valid. In particular, this condition is satisfied if $(\Omega ,\mathfrak {S})$ is a Blackwell space and $({T_1}, \ldots ,{T_n}):\Omega \to [{a_1},{b_1}] \times \cdots \times [{a_n},{b_n}]$ is in addition one-to-one.