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Started reading Diffusions, Markov Processes, and Martingales: Volume 1, Foundations (Cambridge Mathematical Library) and at the beginning of the section titled "Some Frequently Used Notation" I see there is a set defined to be equal to the interval from 0 to infinity, exclusive, shown as the real numbers symbol with a double plus superscript:

From context, $\Bbb{R}^+$ is non-negative real numbers, and $\Bbb{R}^{++}$ is positive real numbers. However, I'd never seen this notation before and was wondering

(a) Is this standard notation?
(b) Is the above is how it is actually to be interpreted?

Thank you!

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    $\begingroup$ I've certainly never seen that notation before. Note that "frequently used" is perfectly valid as a local term, i.e. frequently used within the text itself. $\endgroup$ Commented Aug 16, 2021 at 3:30
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    $\begingroup$ No, it's not standard, and yes, you have interpreted it correctly. Next! $\endgroup$ Commented Aug 16, 2021 at 4:47
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    $\begingroup$ Most math writing uses $\Bbb R^+$ for $(0,\infty)$ & does not use $:=$ for anything. In computer coding $:=$ and $=$ usually have very different meanings from each other. In math $:=$ or $=^{def}$ is a polite way to tell the reader that an equality is a definition $\endgroup$ Commented Aug 16, 2021 at 6:38

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As already mentioned in the comments: (a) no, (b) yes.

If you are looking for standard notations, the ISO 80000-2:2019 (Quantities and units — Part 2: Mathematics) mentions the following remarks when defining the set of real numbers:

$\mathbb{R}^* = \{x \in \mathbb{R} \mid x \neq 0\}$
Other restrictions can be indicated in an obvious way, as shown below. $\mathbb{R}_{> \ 0} = \{x \in R \mid x > 0\}$

The 2009 version was mentioning this example of restriction instead:

$\mathbb{R}_{\geq \ 0} = \{x \in R \mid x \geq 0\}$

Surprisingly, the standard does not mention the notation $\mathbb{R}^+$.

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    $\begingroup$ That ISO document is "for use in the natural sciences and technology"; its notation often disagrees with usage in mathematics itself. $\endgroup$
    – GEdgar
    Commented Oct 25, 2022 at 13:37
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    $\begingroup$ The abstract says "This document is intended mainly for use in the natural sciences and technology, but also applies to other areas where mathematics is used." This is the only ISO standard about mathematical symbols to my knowledge, that's why I consider it as a reference. Is there another international standard? $\endgroup$
    – James
    Commented Oct 25, 2022 at 13:45

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