Let be a measure-preserving system – a probability space equipped with a measure-preserving translation (which for simplicity of discussion we shall assume to be invertible). We will informally think of two points in this space as being “close” if for some that is not too large; this allows one to distinguish between “local” structure at a point (in which one only looks at nearby points for moderately large ) and “global” structure (in which one looks at the entire space ). The local/global distinction is also known as the time-averaged/space-averaged distinction in ergodic theory.

A measure-preserving system is said to be ergodic if all the invariant sets are either zero measure or full measure. An equivalent form of this statement is that any measurable function which is *locally essentially constant* in the sense that for -almost every , is necessarily *globally essentially constant* in the sense that there is a constant such that for -almost every . A basic consequence of ergodicity is the mean ergodic theorem: if , then the averages converge in norm to the mean . (The mean ergodic theorem also applies to other spaces with , though it is usually proven first in the Hilbert space .) Informally: in ergodic systems, time averages are asymptotically equal to space averages. Specialising to the case of indicator functions, this implies in particular that converges to for any measurable set .

In this short note I would like to use the mean ergodic theorem to show that ergodic systems also have the property that “somewhat locally constant” functions are necessarily “somewhat globally constant”; this is not a deep observation, and probably already in the literature, but I found it a cute statement that I had not previously seen. More precisely:

Corollary 1Let be an ergodic measure-preserving system, and let be measurable. Suppose that

for some . Then there exists a constant such that for in a set of measure at least .

Informally: if is locally constant on pairs at least of the time, then is globally constant at least of the time. Of course the claim fails if the ergodicity hypothesis is dropped, as one can simply take to be an invariant function that is not essentially constant, such as the indicator function of an invariant set of intermediate measure. This corollary can be viewed as a manifestation of the general principle that ergodic systems have the same “global” (or “space-averaged”) behaviour as “local” (or “time-averaged”) behaviour, in contrast to non-ergodic systems in which local properties do not automatically transfer over to their global counterparts.

*Proof:* By composing with (say) the arctangent function, we may assume without loss of generality that is bounded. Let , and partition as , where is the level set

For each , only finitely many of the are non-empty. By (1), one has

Using the ergodic theorem, we conclude that

On the other hand, . Thus there exists such that , thus

By the Bolzano-Weierstrass theorem, we may pass to a subsequence where converges to a limit , then we have

for infinitely many , and hence

The claim follows.

## 11 comments

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17 July, 2018 at 7:05 pm

burrobertThanks Terry. Short and sweet. Nice result and demonstration of some basic analysis. The $\varepsilon$ and $2^{-k}$ correspond to the same object. I think $c – 2^{-k}$ in the second last equation could be replaced by $c$.

[Corrected, thanks – T.]18 July, 2018 at 7:39 am

Aritra BhattacharyaIn the first line of proof, do you mean will be bounded?

[Corrected to arctangent – T.]20 July, 2018 at 7:23 am

Maths studentDear Prof. Tao,

I think only finitely many of the sets have nonzero measure; we may put in points with zero measure without changing anything.

Did you ever intend to put all your ergodic lectures somehow in one file, so that it’s easily printable??? I’d be VERY interested.

20 July, 2018 at 7:27 am

Maths studentI’d be able to extract the code by a clever S&R program if the math weren’t images, hence the need…

20 July, 2018 at 11:52 pm

Maths studentI take the correction back, we do have boundedness.

21 July, 2018 at 12:02 am

Maths studentBut still, it’s the and not the .

[Corrected, thanks – T.]20 July, 2018 at 10:33 am

Maths studentAh it’s in the Poincaré legacies book.

22 July, 2018 at 11:49 pm

AnonymousIt seems that ergodicity (similarly to analyticity or even convexity) has the potential to “propagate” certain local properties into global ones.

24 August, 2018 at 1:07 am

Rogier BrusseeI think you want to compose with the arctangent rather than the tangent function.

[Huh, I thought I had fixed that already. In any case, fixed again – T.]5 December, 2018 at 3:02 pm

Fourier uniformity of bounded multiplicative functions in short intervals on average | What's new[…] that ths quantity is essentially constant in (cf. the application of the ergodic theorem in this previous blog post), thus we now […]

22 June, 2019 at 11:06 am

majdoddinIt seems to me that Corollary 1 can be stated more generally:

Let be an ergodic measure-preserving system, and let be measurable. Suppose that

for some . Then there exists a set of measure at least such that .

With Corollary 1 as a special case, and a very similar proof.