清华测度论的讲义2勒贝格测度.pdf

§1 Æ  Æ   §1.1 1 F , F 1 2 m∗ (F F ) = m∗ (F ) + m∗ (F ), 1 2 1 2 ◭ δ := d(F , F )/2 0 F F 1 2 1 2 δ F1 F ε 0 F F {I } n|I | δ 2 1 2 n n m∗ (F F ) |I | − ε. (1) 1 2 n n≥1 {In } I 1 := {I1,n} I2 := {I2,n}I 1 F1 I2 F2 ∗ ∗ m (F ) ≤ |I |, m (F ) ≤ |I |, 1 1,n 2 2,n n≥1 n≥1 m∗ (F ) + m∗ (F ) ≤ |I | + |I | = |I |, (2) 1 2 1,n 2,n n n≥1 n≥1 n≥1 1 12 m∗ (F ) + m∗ (F ) ≤ m∗ (F F ) + ε, 1 2 1 2 ε m∗ (F ) + m∗ (F ) ≤ m∗ (F F ) (3) 1 2 1 2  m∗ (F ) + m∗ (F ) ≥ m∗ (F ∪ F ) 1 2 1 2 (3) ◮ 1 F , F , , F 1 2 n n n m∗ ( F ) = m∗ (F ). i i i=1 i=1