1.4. Lebesgue Measure

Definition(Lebesgue Stieltjes Measure)

A Lebesgue Stieltjes Measure on \(\mathbb{R}\) is a measure on \(\mathcal{R}\) that gives finite measure to each bounded interval.

  

Definition(Generalized Distribution Function)

A function \(F:\mathbb{R}\rightarrow \mathbb{R}\) will be called a generalized distribution function if it is nondecreasing and right continuous.

  

Theorem

If two g.d.fs differ by a constant, i.e., \(F_1(x)=F_2(x)+c\) for \(x\in \mathbb{R}\) and for a constant \(c\), then there is a one-to-one correspondence between the g.d.fs and Lebesgue-Stieltjes measures on \(\mathbb{R}\).

  

Theorem(correspondence from \(\mu\) to \(F\))

Suppose that \(\mu\) is a Lebesgue-Stieltjes measure on \(\mathbb{R}\). Then there exists a generalized distribution function \(F\) satisfying \(F(b)-F(a)=\mu((a,b])\) for all \(\infty<a\le b<\infty\). Moreover the function \(F\) is unique up to an additive constant.

  

Theorem(correspondence from \(F\) to \(\mu\))

If F is a generalized distribution function, then there is a unique Lebesgue-Stieltjes measure \(\mu\) on \((\mathbb{R},\mathcal{R})\) satisfying \(\mu((a,b])=F(b)-F(a)\) for all \(\infty<a\le b<\infty\).

 

• 결론은, 주어진 Distribution Function \(F\)에 대해 \(F(b)-F(a)=\mu((a,b])\)를 만족하는 Unique 한 measure 반드시 존재한다는 것이다. 또한 어떠한 measurable space에서 주어진 measure \(\mu\)에 \(F(b)-F(a)=\mu((a,b])\)를 만족하는 Unique한 Distribution Function \(F\)가 반드시 존재한다는 것이다.

  

Definition(Lebesgue Measure)

A Lebesgue Measure on \(\mathbb{R}\), denoted \(\lambda\) is defined to be the unique measure on \(\mathcal{R}\) assigning to each interval its length as measure s.t

\[ \lambda((a,b])=b-a \mbox{ for all } -\infty<a\le b< \infty. \]   

Remark(Lebesgue Measure on an Interval)

• If \(I\) is any interval, then \(I\in \mathbb{R}\) and \(\mathcal{R}\cap I\) consisting of all Borel sets containes in \(I\), is a \(\sigma\)-field on \(I\), called the Borel \(\sigma\)-field on \(I\), and denoted \(\mathcal{B}(I)\).

• If we restrict \(\lambda\) to \(\mathcal{B}(I)\), then (\(I, \mathcal{B}(I), \lambda\)) is a \(\sigma\)-finite measure space, and if I is a bounded interval, then it is a finite measure space.

• We will often refer to the important example of the unit interval under Lebesgue measure, \(((0,1], \mathcal{B}, \lambda)\), with \(\mathcal{B}=\mathcal{B}((0,1])\). Note that this is a probability space, because \(\lambda((0,1])=1\). (많이 쓰이는 예제)