“Lebesgue measure” on $\textbf {R}^ \infty$

Abstract:

We construct a translation invariant Borel measure $\lambda$ on ${{\mathbf {R}}^\infty } = \prod _{i = 1}^\infty {\mathbf {R}}$ such that for any infinite-dimensional rectangle $R = \prod _{i = 1}^\infty ({a_i},{b_i}), - \infty < {a_i} \leq {b_i} < + \infty$, if $0 \leq \prod _{i = 1}^\infty ({b_i} - {a_i}) < + \infty$, then $\lambda (R) = \prod _{i = 1}^\infty ({b_i} - {a_i})$. Because ${{\mathbf {R}}^\infty }$ is an infinite-dimensional locally convex topological vector space, the measure $\lambda$ can not be $\sigma$-finite.