I am reading Steven E. Shreve's book, titled "Stochastic Calculus for Finance II". I have a query w.r.t. an example given in the book which is as follows:-

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    $\begingroup$ It’s just (arbitrary) notation: $\mathbb P(\text d\omega)=\text d \mathbb P(\omega)$. You can use whatever you prefer. For nice functions, Lebesgue integrals coincide with Riemann integrals and $\ell(\text dx)=\text d\ell(x)=\text dx$. $\endgroup$ – Kevin Apr 13 at 10:36
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    $\begingroup$ It is easily deduced from the additive property of measure: say w and $dw$ are disjoint, then $P\left[w\cup dw\right]= P\left[w\right]+ P\left[dw\right]$, which you can rearrange to get: $dP\left[w\right]=P\left[dw\right]$ $\endgroup$ – Magic is in the chain Apr 13 at 17:29
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    $\begingroup$ When you made the lebesgue measure analogy using $b$ and $a$, you showed that the probability of the interval was equal to its length. In the same way, P(dw) = dw because dw is the length of that interval. $\endgroup$ – mark leeds Apr 13 at 19:39

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