Probability measure annihilating all finite-dimensional subspaces

2025/12/25 by Nizar El Idrissi, Idrissi, Nizar El, Hicham Zoubeir +1
Mathematics · #advanced mathematical theories #Advanced Banach Space Theory #Advanced Topology and Set Theory

paper · doi:10.48550/arxiv.2512.21664

Abstract

We propose in this short note a prime numbers-based method for constructing probability measures on infinite-dimensional Banach spaces annihilating all finite-dimensional subspaces, supplementing the methods of construction of Gaussian measures and infinite-product-type probability measures. This new method confirms that probability measures with this property are generic amongst probability measures that are not supported on finite-dimensional subspaces. In the process, we show the existence of an uncountable measurable family of independent vectors having the cardinality of the continuum in any infinite-dimensional Banach space.

Citations

• Almost sure linear independence of absolutely continuous Hilbert space-valued random vectors with respect to a special class of Hilbert space probability measures
• Independence, infinite dimension, and operators

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