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--- math105-s22:notes:lecture_12 [2022/02/23 23:58]
pzhou [Pugh 6.8: Vitali Covering]
+++ math105-s22:notes:lecture_12 [2022/02/25 00:35] (current)
pzhou
@@ Line 32: / Line 32: @@
  $\int F_+(x) dx = \int F_- (x) dx$
 hence  $F_+(x) = F_-(x)$  for almost all  $x$ . Thus, for a.e.  $x$ , we have $\uint f(x,y) dy = \lint f(x,y) dy$, thus  $\int f(x,y) dy$  exists for a.e. x.
+==== A Lemma ====
+Suppose  $A$  is measurable, and  $B \In A$  any subset, with  $B^c = A \RM B$ . Then
+ $m(A) = m^*(B) + m_*(B^c)$
+Proof:
+ $m^*(B) = \inf \{ m(C) \mid A \supset C \supset B, C \text{measurable} \} = \inf \{ m(A) - m(C^c) \mid A \supset C \supset B, C \text{measurable} \}$
+ $= m(A) - \sup \{ m(C^c) \mid A \supset C \supset B, C \text{measurable} \} = m(A) - \sup \{ m(C^c) \mid C^c \subset B^c, C^c \text{measurable} \} = m(A) - m_*(B^c)$
 ===== Pugh 6.8: Vitali Covering =====

Lecture Notes