現代数学の系譜工学物理雑談古典ガロア理論も読む49

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50: １３２人目の素数さん [sage] 2017/12/29(金) 01:04:58.18 ID:gcYWyS10

>>44
＞というが、それ（”全体で”）を導くことは、定理1.7（「f はある開区間の上でリプシッツ連続である」）をいうだけなら、不必要では？
＞（　上記のある区間（c,d）で、リプシッツ連続を言えば、定理1.7の証明は、そこで終わってないかい？ ）

その言い分そのものは正しいが、そのような (c,d) を見つける方法が全く自明ではなく、
ベールのカテゴリ定理を使わなければ そういう (c,d) が出て来ない、という話をしているんだよ。
つまり、(a,b) ⊂ B_f という条件に限定しても、例の定理の証明は ちっとも自明になってないってこと。
少し詳しく見てみようか？まず前提として、

状況Ａ：
――――――――――――――――――――――――――――――――――――――
「 (a,b) ⊂ B_f を満たす開区間(a, b)が存在する」という条件からは
「 f は(a,b)上の　全　体　で　リプシッツ連続である」という条件は導けない
――――――――――――――――――――――――――――――――――――――

という「状況Ａ」があるわけだ。すると、次のようになる。

(a,b)⊂B_f が成り立つとする。このとき、状況Ａにより、f は(a,b)全体でリプシッツ連続であるとは断言できない。
しょうがないので、(a,b)内の十分小さな区間(c,d)を取る。当然ながら、(c,d)⊂B_f である。
では、f は(c,d)上でリプシッツ連続なのか？残念ながら、状況Ａを区間(c,d)に対して適用すれば、
f は(c,d)上でリプシッツ連続だとは断言できない。では、(c,d)内の更なる小さな区間(s,t)を考えたらどうか？
当然ながら、(s,t)⊂B_f である。では、f は(s,t)上でリプシッツ連続なのか？
残念ながら、「状況Ａ」を区間(s,t)に対して適用すれば、f は(s,t)上でリプシッツ連続だとは断言できない。

……このように、いくら小さな開区間に限定しても、状況Ａがその開区間に適用できるので、
f がその区間の上でリプシッツ連続であるとは断言できなくなってしまう。
では、どうやって望みの部分区間(c,d)を見つければいいのか？
そのために本当に必要になるのが、ベールのカテゴリ定理である。
結局、(a,b) ⊂ B_f という条件に限定しても、例の定理の証明は ちっとも自明にならないのである。

http://rio2016.5ch.net/test/read.cgi/math/1514376850/50

67: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [sage] 2018/01/01(月) 17:07:53.69 ID:dCRrvhl7

で、勝手ながら、年末年始に読んだ関連を貼るよ（＾＾

まず、関連参考：検索でヒットしたので貼る。
BaireCategory.pdfの”3. Pointwise limits of continuous functions.”に、「422に書いた定理」の関連記述
「Theorem. If f : R → R is a pointwise limit of continuous functions,
then Df is Fσ meager (that is, a countable union of closed sets with empty interior).
(In particular, by Baire's theorem, f is continuous on a dense subset of R.)」とあり（当たり前か？ （＾＾ ）

http://www.math.utk.edu/~freire/teaching/m447f16/m447f16index.html
MATH 447- Advanced Calculus I- Fall 2016- A. FREIRE
(or: ANALYSIS IN R^n)
(抜粋)
http://www.math.utk.edu/~freire/teaching/m447f16/BaireCategory.pdf
Sets of discontinuity and Baire's theorem Baire Category Notes (5 problems) (the problems are HW8, due Friday 11/4)A. FREIRE 2016
(抜粋)
1. Sets of discontinuity. For f : R → R, we define
Df = {x ∈ R; f is not continuous at xg:

3. Pointwise limits of continuous functions.
Theorem. If f : R → R is a pointwise limit of continuous functions,
then Df is Fσ meager (that is, a countable union of closed sets with empty interior).
(In particular, by Baire's theorem, f is continuous on a dense subset of R.)

Proof. We know Df = ∪ n>=1 D1/n (see Section 1), so it suffices to show
that the closed sets Dε have empty interior, for any ε > 0.
By contradiction, suppose Dε contains an open interval I.
We'll find an open interval J ⊂ I disjoint from Dε!
Let fn → f pointwise on R, with each fn : R → R continuous.
For each N >= 1, consider the set:
CN = {x ∈ I; (∀m, n >= N)|fm(x) - fn(x)| <= ε/3}.
Clearly ∪ N>=1 CN = I (by pointwise convergence). QED
(引用終り)

つづく

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68: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [sage] 2018/01/01(月) 17:08:32.82 ID:dCRrvhl7

>>67 つづき

（上記の関連参考：出典URL）
http://www.math.utk.edu/~freire/teaching/m447f16/m447f16topics.html
Math 447 Fall 2016- A. FREIRE
TOPICS
PART I: Topology

Supplementary handouts (for advanced students):
(adapted from more advanced classes and not yet in final form)

http://www.math.utk.edu/~freire/teaching/m447f16/GeneralTopologyReview.pdf
Definitions and Theorems from General Topology

http://www.math.utk.edu/~freire/teaching/m447f16/BanachSpace.pdf
Locally compact Banach spaces are finite dimensional (includes 4 problems)

http://www.math.utk.edu/~freire/teaching/m447f16/SpacesOfContinuousFunctions.pdf
Spaces of Continuous Functions (outdated)

http://www.math.utk.edu/~freire/teaching/m447f16/StoneWeierstrassNotes.pdf
Stone-Weierstrass theorem-notes (includes 6 problems)

http://www.math.utk.edu/~freire/teaching/m447f16/AscoliArzelaNotes.pdf
Ascoli-Arzela-Notes (final-included 7 exercises with solutions, and 11 extra problems.)

http://www.math.utk.edu/~freire/teaching/
Alex Freire
Department of Mathematics
University of Tennessee
（終り）

つづく

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70: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [sage] 2018/01/01(月) 17:10:50.85 ID:dCRrvhl7

>>69 つづき

http://www.math.wvu.edu/~kcies/prepF/BookIdensity/BookIdensity.pdf
I-DENSITY CONTINUOUS FUNCTIONS Krzysztof Ciesielski他 1994- 被引用数: 84
(抜粋)
CHAPTER 1
The Ordinary Density Topology
1.1. A Simple Category Topology

To gain some insight into what is happening with limits like this, it is useful
to generalize this idea to a topological setting.
A nonempty family J ⊂P(X) of subsets of X is an ideal on X if A ⊂ B and
B ∈ J imply that A ∈ J and if A∪B ∈ J provided A,B ∈ J. An ideal J on X
is said to be a σ-ideal on X if ∪n∈N An ∈ J for every family {An : n ∈ N} ⊂ J.
Let J be an ideal on R and Ｔo be the ordinary topology on R. The set
T (J) = {G ＼ J : G ∈ Ｔo, J ∈ J}
is a topology on R which is finer than Ｔo. The following proposition is evident
from the definitions.

Proposition 1.1.1. Let J be a σ-ideal on R and T (J ) be as above. For
f : (R, T (J )) → (R, Ｔo) and x0 ∈ R the following statements are equivalent to
each other.
(i): f is continuous at x0.
(ii): Given ε > 0 there is a δ > 0 such that
{x ∈ (x0 − δ, x0 + δ) : |f(x) − f(x0)| ≧ ε} ∈ J.

つづく

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71: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [sage] 2018/01/01(月) 17:11:20.87 ID:dCRrvhl7

>>70 つづき

Theorem 1.1.4. Let J be a σ-ideal and f : R → R. The following statements
are equivalent.
(i): The function f is J -continuous J -a.e.
(ii): There exists a set K ∈ J such that the restricted function f|Kc is continuous.

Furthermore, if the ideal J contains no interval, then the following statement is
equivalent to (i) and (ii)
(iii): There exists a function g : R → R such that f = g J -a.e. and g is continuous in the ordinary sense J -a.e.

Proof. The fact that (ii) implies (i) is obvious. Suppose (i) is true and let
f be J -continuous on a set M = Jc for J ∈ J. For each n ∈ N and x ∈ M,
by Proposition 1.1.1(ii) there is an open interval I(n, x) and a J(n, x) ∈ J such
that
x ∈ I(n, x) \ J(n, x) ⊂ f−1((f(x) − 1/n, f(x) + 1/n)).
For each fixed n, there must be a countable sequence xn,m ∈ M such that
M ⊂∪m∈N I(n, xn,m).
Let
K = J ∪ ∪ n,m∈N J(n, xn,m) ∈ J.
If x ∈ Kc and ε > 0, then there must exist natural numbers n and m such that
2/n < ε and x ∈ I(n, xn,m). Then |f(x) − f(xn,m)| < 1/n so that
f(x) ∈ (f(xn,m) − 1/n, f(xn,m) + 1/n) ⊂ (f(x) − ε, f(x) + ε)

つづく

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72: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [sage] 2018/01/01(月) 17:11:49.95 ID:dCRrvhl7

>>71 つづき

and
I(n, xn,m) ∩ Kc ⊂ f−1((f(x) − ε, f(x) + ε)).
Hence, f|Kc is continuous at x.

To prove the last part of the theorem, note first that (iii) implies (ii) even
without the restriction that J contains no interval. Now suppose that J contains
no interval and that f,K are as in (ii). Define
(1) G(x) = lim sup t→x,t∈Kc f(t)
and
(2) g(x) = G(x) when G(x) is finite,
　 or = f(x) otherwise.
In particular, it follows from (ii) that f|Kc = g|Kc . Let x ∈ Kc and ε > 0.
According to (ii) there is a δ > 0 such that
(3) |g(y) − g(x)| = |f(y) − f(x)| < ε/2
whenever y ∈ (x − δ, x + δ) ∩Kc. If z ∈ (x − δ, x + δ) ∩K, then the assumption
that K can contain no nonempty open set implies the existence of a sequence
{zn : n ∈ N} ⊂ (x − δ, x + δ) ∩ Kc
such that f(zn) → G(z). Hence, by (3), G(z) is finite, so g(z) = G(z) and
|g(z) − g(x)| ? ε/2 < ε. Therefore, g is continuous at x. QED

つづく

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74: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [sage] 2018/01/01(月) 17:12:44.18 ID:dCRrvhl7

>>73 つづき

If condition (i) in Theorem 1.1.4 is strengthened to everywhere, the following corollary results.

Corollary 1.1.8. Let J be a σ-ideal which contains no nonempty open set.
A function f : R → R is continuous everywhere if, and only if, it is J -continuous everywhere.

Proof. If f is continuous, then it is clearly J -continuous. So, suppose f is
J -continuous everywhere, x0 ∈ R and ε > 0. Using Proposition 1.1.1(ii), there
must be an ordinary open neighborhood G0 of x0 such that
F0 = {x ∈ G0 : |f(x) − f(x0)| > ε} ∈ J.
Suppose there is an x1 ∈ F0. Choose δ > 0 such that
δ < |f(x1) − f(x0)| − ε.
As before, there exists an ordinary open neighborhood G1 ⊂ G0 of x1 such that
F1 = {x ∈ G1 : |f(x1) − f(x)| > δ} ∈ J.
It is clear that G1 ⊂ F0 ∪ F1 ∈ J, because |f(x1) − f(x0)| > ε + δ. But, this
implies J contains a nonempty open set, which contradicts the condition placed
on J in the statement of the corollary. This contradiction shows that F0 = Φ.
The preceding corollary demonstrates that global J -continuity may not be a
very useful concept. In particular, it is worthwhile noting for future reference
that global I-continuity and global N-continuity are no different than ordinary continuity.

(引用終り)

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76: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [sage] 2018/01/01(月) 17:13:34.44 ID:dCRrvhl7

>>74 つづき

（上記の関連参考：出典URL）
http://www.math.wvu.edu/~kcies/
Krzysztof Chris Ciesielski, Ph.D. Professor of Mathematics at Department of Mathematics, West Virginia University and Adjunct Professor at Medical Image Processing Group, Dept. of Radiology, Univ. of Pennsylvania.
(抜粋)
Books:
(with L. Larson and K. Ostaszewski) I-density continuous functions, Memoirs of the AMS vol. 107 no 515, 1994; MR 94f:54035.
(引用終り)

https://www.amazon.co.jp/I-Density-Continuous-Functions-American-Mathematical/dp/0821825798
I-Density Continuous Functions (Memoirs of the American Mathematical Society) (英語) Krzysztof Ciesielski (著),? Lee Larson (著),? Krzysztof Ostaszewski (著) 1994/1/1

http://www.jstor.org/stable/44151978?seq=1#page_scan_tab_contents
JOURNAL ARTICLE I-density Continuous Functions Krzysztof Ciesielski, Lee Larson and Krzysztof Ostaszewski Real Analysis Exchange Vol. 15, No. 1 (1989-90), pp. 13-15 Published by: Michigan State University Press
（終り）

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