如果一个集合的Lebesgue测度为0, 那么它的自己也是Lebesgue可测的并且其测度也为0.

Theorem

A subset of a zero(0) Lebesgue measure is Lebesgue measurable.

proof

We will use the Carathéodory’s criterion to prove this statement.

1. Let $m(E)=0$, where $m$ is the Lebesgue measure.

2. Let $A$ be any set, and $B\subset E$. Thus

   $m^{\ast }(A)⩽m^{\ast }(A\cap B)+m^{\ast }(A\cap B^C)$

   where $m^{\ast }$ is the Lebesgue outer measure (Lebesgue outer measure is countably subadditive.)

3. Since the Lebesgue outer measure is monotone, we have

   $$0⩽m^{\ast }(A\cap B)⩽m^{\ast }(B)⩽m^{\ast }(E)=0$$

   Hence, we have

   $$m^{\ast }(A)⩽m^{\ast }(A\cap B^C)$$

4. Since $A\cap B^C\subset A$, we have

   $$m^{\ast }(A\cap B^C)⩽m^{\ast }(A)$$

5. By 3 and 4, we have

   $$m^{\ast }(A)=m^{\ast }(A\cap B)+m^{\ast }(A\cap B^C)$$.

6. By the Carathéodory’s criterion, we have proved that $B$ is Lebesgue measurable.

7. $0⩽m(B)⩽m(E)=0\to m(B)=0$

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