“Lebesgue measure” on 𝑅^{∞}

Abstract

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