μ-distributions and μ-distributed sequences

Author

Noah Graham Giddings

Date of Award

8-2025

Degree Type

Thesis

Degree Name

Honors Thesis

Department

Mathematics

First Advisor

John Herr

Second Advisor

William W. Johnston

Abstract

The project attempts to generalize the notion of a uniform distribution for a broader class of probability measures and illustrate one construction of a sequence that satisfies these properties. A sequence ⟨a_n⟩ is uniformly distributed if the limiting relative frequency of sequence elements in any interval I ⊆ [0,1] corresponds with length(I). We generalize the notion of “uniform distribution” for an atomless Borel probability measure µ. We say that a sequence ⟨a_n⟩ is µ-distributed if the limiting relative frequency of sequence elements in any interval I ⊆ [0,1] equals µ(I). Given some atomless Borel probability measure µ, we provide a construction of a µ-distributed sequence ⟨A_n⟩ and adapt basic results for uniform distributions to µ-distributions. The invitation to demonstrate ⟨A_n⟩ is computable remains open for further work.

Recommended Citation

Giddings, Noah Graham, "μ-distributions and μ-distributed sequences" (2025). Undergraduate Honors Thesis Collection. 811.
https://digitalcommons.butler.edu/ugtheses/811