Lebesgue積分講義ノート

4.2. 基本性質🔗

Proposition4.2.1
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任意の A,B \subset \RR^d に対して次が成り立つ．

  • 0 \le \lambda_*(A) \le \lambda^*(A)

  • A \subset B ならば \lambda_*(A) \le \lambda_*(B)

Lean code for Proposition4.2.1●3 theorems
  • theoremdefined in NoteKsk/«04lebesgue-inner».lean
    complete
    theorem NoteKsk.Chapter04.lambdaInner_nonneg {d : ℕ}
  (A : Set (NoteKsk.Space d)) : 0 ≤ NoteKsk.Chapter04.lambdaInner A
    theorem NoteKsk.Chapter04.lambdaInner_nonneg
  {d : ℕ} (A : Set (NoteKsk.Space d)) :
  0 ≤ NoteKsk.Chapter04.lambdaInner A
  • theoremdefined in NoteKsk/«04lebesgue-inner».lean
    complete
    theorem NoteKsk.Chapter04.lambdaInner_le_lambdaStar {d : ℕ}
  (A : Set (NoteKsk.Space d)) :
  NoteKsk.Chapter04.lambdaInner A ≤ NoteKsk.Chapter03.lambdaStar A
    theorem NoteKsk.Chapter04.lambdaInner_le_lambdaStar
  {d : ℕ} (A : Set (NoteKsk.Space d)) :
  NoteKsk.Chapter04.lambdaInner A ≤
    NoteKsk.Chapter03.lambdaStar A
  • theoremdefined in NoteKsk/«04lebesgue-inner».lean
    complete
    theorem NoteKsk.Chapter04.lambdaInner_mono {d : ℕ} {A B : Set (NoteKsk.Space d)}
  (hAB : A ⊆ B) :
  NoteKsk.Chapter04.lambdaInner A ≤ NoteKsk.Chapter04.lambdaInner B
    theorem NoteKsk.Chapter04.lambdaInner_mono {d : ℕ}
  {A B : Set (NoteKsk.Space d)}
  (hAB : A ⊆ B) :
  NoteKsk.Chapter04.lambdaInner A ≤
    NoteKsk.Chapter04.lambdaInner B
Proof for Proposition 4.2.1
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  • コンパクト集合 K \subset A に対して，外測度の単調性より

\lambda^*(K) \le \lambda^*(A)

である． 左辺について上限を取れば

\lambda_*(A) \le \lambda^*(A)

を得る． また外測度は常に非負なので，その上限である \lambda_*(A) も非負である．

  • A \subset B とする． K \subset A なる任意のコンパクト集合は，そのまま K \subset B も満たす． したがって，A に対して取る上限は B に対して取る上限以下である．

Proposition4.2.2
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Theorem 4.4.1
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コンパクト集合では内外が一致する. コンパクト集合 K \subset \RR^d に対して

\lambda_*(K)=\lambda^*(K)

が成り立つ．

Lean code for Proposition4.2.2●2 theorems
  • theoremdefined in NoteKsk/«04lebesgue-inner».lean
    complete
    theorem NoteKsk.Chapter04.lambdaInner_eq_lambdaStar_of_isCompact {d : ℕ}
  {K : Set (NoteKsk.Space d)} (hK : IsCompact K) :
  NoteKsk.Chapter04.lambdaInner K = NoteKsk.Chapter03.lambdaStar K
    theorem NoteKsk.Chapter04.lambdaInner_eq_lambdaStar_of_isCompact
  {d : ℕ} {K : Set (NoteKsk.Space d)}
  (hK : IsCompact K) :
  NoteKsk.Chapter04.lambdaInner K =
    NoteKsk.Chapter03.lambdaStar K
    Compact sets have equal inner and outer measure.
  • theoremdefined in NoteKsk/«04lebesgue-inner».lean
    complete
    theorem NoteKsk.Chapter04.lambdaInner_eq_zero_of_lambdaStar_eq_zero {d : ℕ}
  {A : Set (NoteKsk.Space d)}
  (hA : NoteKsk.Chapter03.lambdaStar A = 0) :
  NoteKsk.Chapter04.lambdaInner A = 0
    theorem NoteKsk.Chapter04.lambdaInner_eq_zero_of_lambdaStar_eq_zero
  {d : ℕ} {A : Set (NoteKsk.Space d)}
  (hA :
    NoteKsk.Chapter03.lambdaStar A = 0) :
  NoteKsk.Chapter04.lambdaInner A = 0
    A set of outer measure zero has inner measure zero.
Proof for Proposition 4.2.2
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K 自身がコンパクトだから， 内測度の上限を取る集合の中に K そのものが含まれている． したがって

\lambda_*(K) \ge \lambda^*(K)

である． 逆向きの不等式は前命題から従う．

Corollary4.2.3
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外測度零集合の内測度. \lambda^*(A)=0 ならば

\lambda_*(A)=0

である．

Proof for Corollary 4.2.3
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基本性質の (1) より

0 \le \lambda_*(A) \le \lambda^*(A)=0

だからである．