Three random variable transformations giving Heisenberg-type uncertainty relations

Though intrinsically probabilistic, the 1927 quantum mechanics Heisenberg uncertainty relation is still non-existent as a native Kolmogorov probability construct. This paper fills the gap via three random variable transformations that generate Heisenberg-type uncertainty relations on Kolmogorov probability space (Ω,Σ,μ) without resorting to any quantum mechanics notion or formalism. A sufficient condition is given under which a class of Heisenberg uncertainty relations on (Ω,Σ,μ) is a bounded distributive lattice. Some applications in progress of the above results are anticipated.