[過去ﾛｸﾞ] 現代数学の系譜 工学物理雑談 古典ガロア理論も読む49 (658ﾚｽ)

現代数学の系譜 工学物理雑談 古典ガロア理論も読む49 http://rio2016.5ch.net/test/read.cgi/math/1514376850/

上下前次1-新
通常表示 512ﾊﾞｲﾄ分割 ﾚｽ栞
抽出解除 ﾚｽ栞

---

このｽﾚｯﾄﾞは過去ﾛｸﾞ倉庫に格納されています｡
次ｽﾚ検索 歴削→次ｽﾚ 栞削→次ｽﾚ 過去ﾛｸﾞﾒﾆｭｰ

---

73: 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) 17:12:19.36 ID:dCRrvhl7 >>72 つづき The following example is interesting in light of the previous theorem. Example 1.1.5. Let I be the σ-ideal consisting of all first category subsets of R. I-continuity is often called qualitative continuity [26]. It is well-known in this case that f is a Baire function if, and only if, f is qualitatively continuous I-a.e. In particular, combining Example 1.1.5 with Theorem 1.1.4 yields the following well-known corollary, which will be useful in the sequel. Corollary 1.1.6. Let f : R → R. The following statements are equivalent. (i): f is a Baire function. (ii): There exists a residual set K such that f|K is continuous.*2 (iii): f is qualitatively continuous I-a.e. In the case of Lebesgue measure, the following is true. *2 A set is residual if its complement is first category. This is often called comeager. つづく http://rio2016.5ch.net/test/read.cgi/math/1514376850/73

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) つづく http://rio2016.5ch.net/test/read.cgi/math/1514376850/74

---

ﾒﾓ帳

(0/65535文字)

---

上下前次1-新書関写板覧索設栞歴

---

ｽﾚ情報 赤ﾚｽ抽出 画像ﾚｽ抽出 歴の未読ｽﾚ

---

Google検索

Wikipedia

---

ぬこの手 ぬこTOP 0.025s