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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."

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    $\begingroup$ When two measures on the same sigma-algebra give the same numerical values to every sets in that sigma-algebra, they're simply the same measure. $\endgroup$ – BigbearZzz Sep 22 '16 at 20:13
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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.

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