# Continuous transformations of probability measures and their transport representations

arXiv: https://arxiv.org/abs/2604.16653 · HTML: https://mlsanity.com/paper/2604.16653/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Apr 2026.

## Built on by (most cited first, 4 of 4)

- [Measure-to-measure Regression with Transformers](https://mlsanity.com/paper/2605.28075/) (2026)
- [Any-Dimensional Learning by Sampling](https://mlsanity.com/paper/2607.07680/) (2026)
- [Universal Approximation of Measure-to-Measure Operators by Pushforwards](https://mlsanity.com/paper/2609.35483/) (2026)
- [Stability of Measure-to-Measure Transformers on Sub-Gaussian Data](https://mlsanity.com/paper/2610.07717/) (2026)

## Abstract (arXiv)

Given a function F transforming a probability measure μ into another one F(μ), we study the existence and regularity of a transport representation of it. That is, we ask whether we can represent the image F(μ) of the input probability measure μ as the push-forward of μ by a map f(·,μ) which may depend on μ; and furthermore, how regular f can be chosen depending on F. Even if F is continuous and a transport representative exists, it cannot necessarily be chosen in a continuous way; however, if F is Lipschitz continuous with respect to the Wasserstein distance, then f can be chosen continuous. We provide several examples to illustrate the sharpness of our assumptions. This question is motivated by approximation results for transformations of probability distributions with transformers.