I have started reading Rick Durrett's "Probability: Theory and Examples" Edition 4.1.
There is a section quite at the beginning that I do not understand. it starts OK but then at the last line "When
F
(
x
) =
x
the resulting measure is called
Lebesgue measure"
Does this imply that when the domain and co-domain has the same value the Stieltjes measure function becomes a Lebesgue Measure? What i have understood so far about Lebesgue Measure doesn't include this definition.
Quote:
"Associated with each Stieltjes measure function
F
there is a unique
measure μ on ( $\mathbb{R}$, $\mathcal{R}$)
with
μ
((
a,b
]) =
F
(
b
)
−
F
(
a
)
When
F
(
x
) =
x
the resulting measure is called
Lebesgue measure."
Best Answer
Using Durrett's notation, a Stieltjes measure function is a function $F : \mathbb{R} \to \mathbb{R}$ that is nondecreasing and right-continuous, whereas $\mu : \mathcal{R} \to \mathbb{R}$ assigns a measure to every Borel subset of $\mathbb{R}$. Every Stieltjes function $F$ defines a $\mu$. $F$ doesn't need to be the identity function, but if it is, then the $\mu$ of Theorem 1.1.2 is called Lebesgue measure.