Is it okay to write $[0,\infty]$?

Question

In my tuition school, a teacher wrote $[0, \infty]$ and i asked him if it is ever possible to write $\infty]$. He sternly replied not to question about silly things and focus on what is being taught. I thought if it was every possible and later asked my physics teacher about it, and he gave me a nice answer... He told that there is a professor in Calcutta University and a student had asked him the similar question i asked, and he had replied that it is not incorrect. I do not know why, but is there a mathematical reasoning, or just an intuition, or cant it be written at all? Which is more accurate(correct): $[0, \infty)$ or $[0,\infty]$?

Answer 1

It sure is, and the meaning is exactly as you'd expect: since it's an interval descriptor that ends with a "$]$", then it means that it includes its right endpoint, which means it includes

$$\infty$$

i.e. $\infty \in [0, \infty]$. Now you may think that weird, because most likely you were taught that "$\infty$" is just some "notational symbol" or something "used with limits" and "not really a 'number'". This is, in fact, half true and half not true. It is a notational symbol alright, but it is also an object that, while not a real number, is nonetheless something else called an "extended real number" and lives in a space called the "extended real number line".

The extended real number line is just the usual reals together with two elements $\infty$ and $-\infty$ added to them:

$$\mathbb{R}_\mathrm{ext} := \mathbb{R} \cup \{ -\infty, \infty \}$$

and we extend the ordering operation $\le$ (a Boolean-valued function) on the real numbers by asserting

$$[-\infty \le x \le \infty] = \mathrm{True}$$

for every extended real number $x$, and $\mathrm{False}$ otherwise (while also retaining the usual order for ordinary reals, of course!). You should check that this also implies $-\infty < \infty$. Intuitively, we "cap" the "ends" of the real number line with the two elements, in the obvious fashion. Now $[0, \infty]$ is a proper extended real interval, but is not a real interval.

And that last point is important - because while this construction is 100% mathematically kosher, you have to be careful when working with $\infty$ and not to mis-apply rules from the reals that don't apply to it. For example, $\infty - \infty$ is undefined, and you can't, thus, subtract $\infty$ from both sides of an expression and cancel it out. This should make sense to you given the usage in limits, where when this form appears, it signals that you have to take a more clever approach to examining your limit problem. Adding $\infty$ into play as a "real" number doesn't change that because it doesn't work like real numbers do. One thus also has to be careful not to confuse things defined in $\mathbb{R}$ with things defined in $\mathbb{R}_\mathrm{ext}$. Paying attention to domains is very important.

Answer 2

This is used in measure theory. A measure is a function that takes subsets of a given set and assigns each one of them a number in $[0, \infty]$. So not only is it allowed, it's encouraged.

And your teacher was wrong for accusing you of asking a silly question.