Typical F_{σ} Sets and Typical Continuous Functions

Shingo SAITO

（斎藤新悟）

University College London

http://www.ucl.ac.uk/~ucahssa/

START Next

Typical Closed Sets

Work in I = [0, 1]. K := {K ⊂ I | K: closed }

equipped with the Hausdorff metric d

(d(K, ∅) := 1 for nonempty K ∈ K).

d defines residuality (comeagreness) on K.

e.g.

{K ∈ K | K: null }: residual in K.

We say typical closed sets are null.

↑ all in some residual set

Typical F_{σ} Sets

We need to define residuality on F_{σ}. Look at the surjection

K^{ℕ} ⟶  F_{σ}
∈  ∈
(K_{n}) ⟼ ⋃_{n=1}^{∞} K_{n}

This map gives F_{σ} a topology, hence residuality. It turns out that for F ⊂ F_{σ},

F: residual ⟺ F ⊃ {F ∈ F_{σ} | F: dense }. This residuality is not useful.

Instead of giving F_{σ} a topology, we define residuality directly:

Def

F ⊂ F_{σ} is residual if

{(K_{n}) ∈ K^{ℕ} | ⋃_{n=1}^{∞} K_{n} ∈ F}

is residual in K^{ℕ}.

Prop

For a σ-ideal I on I, T.F.A.E.: I ∩ F_{σ} is residual in F_{σ}

(i.e. F ∈ I for typical F ∈ F_{σ}); I ∩ K is residual in K

(i.e. K ∈ I for typical closed K).

Typical continuous functions

C(I) := {f: I ⟶ ℝ | f: continuous }

equipped with the supremum norm. C(I) has residuality.

Our main theorem links typical continuous functions and typical F_{σ} sets.

Def

x ∈ I is a knot point of f ∈ C(I) if

D^{+} f(x) := lim sup_{y↘x} \frac{f(y) − f(x)}{y − x} = ∞,
D_{+} f(x) := lim inf_{y↘x}  = −∞,
D^{−}f(x) := lim sup_{y↗x}  = ∞,
D_{−}f(x) := lim inf_{y↗x}  = −∞.

Back 8(Next)

For f ∈ C(I),

{knot points of f } ∈ G_{δ},
N(f) := {non-knot points of f } ∈ F_{σ}.

Fact (unpublished) For a σ-ideal I on I, T.F.A.E.: N(f) ∈ I for typical f ∈ C(I); I ∩ K is residual in K

(i.e. K ∈ I for typical closed K).

⟺ N(f) ∈ I ∩ F_{σ} for typical f ∈ C(I) ;
⟺ I ∩ F_{σ} is residual in F_{σ} .

Back 9(Next)

Main Thm

For F ⊂ F_{σ}, T.F.A.E.: N(f) ∈ F for typical f ∈ C(I); F is residual in F_{σ}

(i.e. F ∈ F for typical F ∈ F_{σ}).

Proof of Main Thm

Lem

We may find a ‘good’ 𝔸 ⊂ K^{ℕ} × C(I) s.t. ((K_{n}), f) ∈ 𝔸 implies ⋃_{n=1}^{∞} K_{n} = N(f); if A ⊂ K^{ℕ} is residual, then for typical f ∈ C(I),

∃(K_{n}) ∈ A ((K_{n}), f) ∈ 𝔸.

The proof of this lemma is very complicated and uses the Banach-Mazur game.

Back 11(Next)

⇒ Suppose F ⊂ F_{σ} is residual. A := {(K_{n}) ∈ K^{ℕ} | ⋃_{n=1}^{∞} K_{n} ∈ F}: residual.

By Lem, for typical f ∈ C(I), ∃(K_{n}) ∈ A ((K_{n}), f) ∈ 𝔸,

∴ N(f) = ⋃_{n=1}^{∞} K_{n} ∈ F.

⇒ Suppose N(f) ∈ F for typical f ∈ C(I). Take a dense G_{δ} set G ⊂ {f ∈ C(I) | N(f) ∈ F}. A := {(K_{n}) ∈ K^{ℕ} | ∃f ∈ G ((K_{n}), f) ∈ 𝔸}.

(K_{n}) ∈ A implies ⋃_{n=1}^{∞} K_{n} = N(f) ∈ F. Thus it suffices to show that A is residual.

A turns out to be analytic (since 𝔸 is ‘good’). ∴ A has the Baire Property. ∴ A is either meagre or residual

(topological 0-1 law).

Suppose A is meagre. Then A^{c} is residual.

By Lem, for typical f ∈ C(I),

∃(K_{n}) ∈ A^{c} ((K_{n}), f) ∈ 𝔸.

Thus

∃f ∈ G ∃(K_{n}) ∈ A^{c} ((K_{n}), f) ∈ 𝔸. This contradicts the definition of A.

Back  END  Bye