## Journal of Symbolic Logic

- J. Symbolic Logic
- Volume 71, Issue 2 (2006), 611-623.

### A constructive view on ergodic theorems

#### Abstract

Let T be a positive L₁-L_{∞} contraction. We
prove that the following statements are equivalent in constructive
mathematics.

- 1. The projection in L₂ on the space of invariant functions exists;
- 2. The sequence (Tⁿ)
_{n ∈ N}Cesáro-converges in the L₂ norm; - 3. The sequence (Tⁿ)
_{n ∈ N}Cesáro-converges almost everywhere.

As a corollary we obtain a constructive ergodic theorem for ergodic measure-preserving transformations. This answers a question posed by Bishop.

#### Article information

**Source**

J. Symbolic Logic Volume 71, Issue 2 (2006), 611-623.

**Dates**

First available in Project Euclid: 2 May 2006

**Permanent link to this document**

http://projecteuclid.org/euclid.jsl/1146620162

**Digital Object Identifier**

doi:10.2178/jsl/1146620162

**Mathematical Reviews number (MathSciNet)**

MR2225897

#### Citation

Spitters, Bas. A constructive view on ergodic theorems. J. Symbolic Logic 71 (2006), no. 2, 611--623. doi:10.2178/jsl/1146620162. http://projecteuclid.org/euclid.jsl/1146620162.