This is again a note on my talk in the Szemerédi’s theorem seminar, going through Furstenberg’s book. In this round, my part is to introduce compact extension.

Let be an abelian group of measure preserving transformations on , be an extension map.

i.e. s.t. sends sets to sets;

**Definition:** A sequence of subsets of is a **Folner sequence** if and for any ,

**Proposition:** For any Folner sequence of , for any , converges weakly to the orthogonal projection of onto the subspace of -invariant functions. (Denoted where .

**Proof:** Let

For all ,

Since is -preserving, is unitary on . Therefore we also have .

For , suppose there is subsequence where converges weakly to some .

By the property that , we have for each , is -invariant. i.e.

However, since hence all are in hence . Therefore ,

**Recall:** 1).

i.e. fibred product w.r.t. the extension map .

2)For ,

**Definition:** A function is said to be **almost periodic** if for all , there exists s.t. for all and almost every ,

**Proposition:** Linear combination of almost periodic functions are almost periodic.

**Proof:** Immediate by taking all possible tuples of for each almost periodic function in the linear combination corresponding to smaller l.

**Definition:** is a **compact extension** if:

**C1: ** , contains a basis of .

**C2: ** The set of almost periodic functions is dense in

**C3: ** For all , there exists s.t. for any and almost every , we have

**C4: ** For all , there exists s.t. for any , there is a set , for all

**C5: ** For all , let where

Let be a Folner sequence, then iff .

**Theorem:** All five definitions are equivalent.

**Proof: “C1 C2″**

Since almost periodic functions are closed under linear combination, it suffice to show any element in a set of basis is approximated arbitrarily well by almost periodic functions.

Let our basis be as given in C1.

For all , the associated linear operator where is bounded. Hence it suffice to check for a dense set of . We consider the set of all fiberwise bounded i.e. for all , .

For all , we perturb by multiplying it by the characteristic function of a set of measure at least to get an almost periodic function.

**“C2 C3″:**

For any , there exists almost periodic, with . Let be the functions obtained from the almost periodicity of with constant , .

Let , since

Hence , has measure at most , therefore .

For all , if then

Hence

If then vanishes on so that .

Hence satisfies C3.

**“C3 C4″:**

This is immediate since for all , we have on hence

. Hence satisfies C4.

**“C4 C5″:**

For all , by C4, there exists s.t. for any , there is a set , for all

W.L.O.G. we may suppose all are bounded since by making slighter larger we can modify the unbounded parts to be bounded.

, suppose .

Recall in C5 we have , and .

For each , we have

Hence we have

Hence for large enough , there exists s.t. is as small as we want.

We may find with s.t. for all and for all , we have

On the other hand, by construction there is with for all , with .

Hence for .

Let we get . Hence C5 holds.

**“C5 C1″**

Let orthogonal to all of such functions. Let be a Folner sequence.

Define

Let be equal to whenever and o.w.

is -invariant is -invariant and bounded.

Therefore , i.e.

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Since , we get

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Hence . For all .

Since is an average of .

for all

Hence . By C5, we obtain . Hence , contain a basis for .

**Definition:** Let be a subgroup of , is said to be **compact relative to ** if the extension is compact.