measure-preserving transformation

保测变换

Dynamical Systems · Graduate · Mathematics Resources editorial compilation (AI-assisted)

Other names / variants: m.p.t. · mpt

Translation evidence: Literature usage

Meaning in context

On a measure space $(X,\mathcal B,\mu)$, a measurable self-map $T$ that satisfies $\mu(T^{-1}A)=\mu(A)$ for all $A\in\mathcal B$.

测度空间上的可测自映射，保持每个可测集合原像的测度。

Example

The doubling map on the circle preserves length although it is not one-to-one.

Usage and distinctions

The transformation need not be invertible; preservation of the image measure of every set is a different assertion.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Omri Sarig — Lecture Notes on Ergodic Theory§1.2, Definition 1.2, measurable and measure-preserving conditions · Checked 2026-10-14

Translation sources

1. 朱子阳 — 算术动力系统：代数数论专题 VI（作者主页讲义）§1，定义1.1：保测变换 · Checked 2026-10-14

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