invariant measure

不变测度

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Translation evidence: Literature usage

Meaning in context

For a measurable T:X→XT:X\to X, a measure μ\mu is invariant if μ(T−1A)=μ(A)\mu(T^{-1}A)=\mu(A) for every measurable set AA.

对可测变换T，每个可测集合的原像与该集合具有相同测度时，μ称为T的不变测度。

Example

Normalized length on the circle is invariant under rotations and under x↦2x mod 1x\mapsto2x\bmod1.

Usage and distinctions

Invariance uses preimages even for noninvertible maps; it does not require every state to be fixed or the measure to be ergodic.

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Definition sources

  1. Omri Sarig — Lecture Notes on Ergodic Theory§1.2 The abstract setup of ergodic theory, Definition 1.2 · 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：T-不变测度 · Checked 2026-10-14
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