## A Summer 2015 Mathematics A To Z: measure

## Measure.

Before painting a room you should spackle the walls. This fills up small holes and cracks. My father is notorious for using enough spackle to appreciably diminish the room’s volume. (So says my mother. My father disagrees.) I put spackle on as if I were paying for it myself, using so little my father has sometimes asked when I’m going to put any on. I’ll get to mathematics in the next paragraph.

One of the natural things to wonder about a set — a collection of things — is how big it is. The “measure” of a set is how we describe how big a set is. If we’re looking at a set that’s a line segment within a longer line, the measure pretty much matches our idea of length. If we’re looking at a shape on the plane, the measure matches our idea of area. A solid in space we expect has a measure that’s like the volume.

We might say the cracks and holes in a wall are as big as the amount of spackle it takes to fill them. Specifically, we mean it’s the least bit of spackle needed to fill them. And similarly we describe the measure of a set in terms of how much it takes to cover it. We even call this “covering”.

We use the tool of “cover sets”. These are sets with a measure — a length, a volume, a hypervolume, whatever — that we know. If we look at regular old normal space, these cover sets are typically circles or spheres or similar nice, round sets. They’re familiar. They’re easy to work with. We don’t have to worry about how to orient them, the way we might if we had square or triangular covering sets. These covering sets can be as small or as large as you need. And we suppose that we have some standard reference. *This* is a covering set with measure 1, *this* with measure 1/2, this with measure 24, this with measure 1/72.04, and so on. (If you want to know what units these measures are in, they’re “units of measure”. What we’re interested in is unchanged whether we measure in “inches” or “square kilometers” or “cubic parsecs” or something else. It’s just longer to say.)

You can imagine this as a game. I give you a set; you try to cover it. You can cover it with circles (or spheres, or whatever fits the space we’re in) that are big, or small, or whatever size you like. You can use as many as you like. You can cover more than just the things in the set I gave you. The only absolute rule is you must not miss anything, even one point, in the set I give you. Find the smallest total area of the covering circles you use. That smallest total area that covers the whole set is the measure of that set.

Generally, measure matches pretty well the intuitive feel we might have for length or area or volume. And the idea extends to things that don’t really have areas. For example, we can study the probability of events by thinking of the space of all possible outcomes of an experiment, like all the ways twenty coins might come up. We find the measure of the set of outcomes we’re interested in, like all the sets that have ten tails. The probability of the outcome we’re interested in is the measure of the set we’re interested in divided by the measure of the set of all possible outcomes. (There’s more work to do to make this quite true. In an advanced probability course we do this work. Please trust me that we could do it if we had to. Also you see why we stride briskly past the discussion of units. What unit would make sense for measuring “the space of all possible outcomes of an experiment” anyway?)

But there are surprises. For example, there’s the Cantor set. The easiest way to make the Cantor set is to start with a line of length 1 — of measure 1 — and take out the middle third. This produces two line segments of length, measure, 1/3 each. Take out the middle third of each of those segments. This leaves four segments each of length 1/9. Take out the middle third of each of those four segments, producing eight segments, and so on. If you do this infinitely many times you’ll create a set that has no measure; it fills no volume, it has no length. And yet you can prove there are just as many points in this set as there are in a real normal space. Somehow merely having a lot of points doesn’t mean they fill space.

Measure is useful not just because it can give us paradoxes like that. We often want to say how big sets, or subsets, of whatever we’re interested in are. And using measure lets us adapt things like calculus to become more powerful. We’re able to say what the integral is for functions that are much more discontinuous, more chopped up, than ones that high school or freshman calculus can treat, for example. The idea of measure takes length and area and such and makes it more abstract, giving it great power and applicability.

## sarcasticgoat 3:31 pm

onMonday, 22 June, 2015 Permalink |Living with black mold on walls, thoughts?

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## Joseph Nebus 4:41 pm

onMonday, 22 June, 2015 Permalink |See if you can get any rent from it. Unfortunately the mold may be aware the only way to get rid of it is to burn the house down and move to another time zone, so it tends to make lowball offers. Make sure you have a flame source and an accelerant in view while negotiating.

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## sarcasticgoat 5:10 pm

onMonday, 22 June, 2015 Permalink |Thanks, I’ll have a chat Head-Spore, the intimidating three inch big mold spore who taunts me as I walk to the bathroom. :)

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## Joseph Nebus 4:49 am

onTuesday, 23 June, 2015 Permalink |Best of luck …

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## howardat58 7:14 pm

onMonday, 22 June, 2015 Permalink |I love the Cantor middle third set. It really is mind blowing on first encounter.

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## John Friedrich 12:29 am

onTuesday, 23 June, 2015 Permalink |Of further interest, the Hausdorff dimension of the Cantor set is ln2/ln3, which proves that it is a fractal.

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## Joseph Nebus 4:56 am

onTuesday, 23 June, 2015 Permalink |Yeah, that’s a neat trait. I might get around to dimensions if I do another a-to-z run, or maybe as an independent discussion.

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## Joseph Nebus 4:55 am

onTuesday, 23 June, 2015 Permalink |It is mind-blowing, and it’s one of those sets that just keeps giving strangeness.

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## Aquileana 2:08 pm

onWednesday, 24 June, 2015 Permalink |5,280 is such an interesting number!… I appreciate that you share all about its twists and meanings with us… Also congratulations on your stats on Twitter, Joseph. All my best wishes. Aquileana :D

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## Joseph Nebus 3:35 am

onThursday, 25 June, 2015 Permalink |I’m glad that you liked. Thank you.

Now on to 6,076! (Feet in a nautical mile. Approximately.)

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## A Summer 2015 Mathematics A To Z: unbounded | nebusresearch 2:55 pm

onFriday, 10 July, 2015 Permalink |[…] might remember the talk about measure, and how it gives an idea of how big a set is. And in that case you might expect an unbounded region […]

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## A Summer 2015 Mathematics A to Z Roundup | nebusresearch 3:03 pm

onFriday, 24 July, 2015 Permalink |[…] Measure. […]

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## Mathematics A to Z: Part 3 | Mean Green Math 11:09 am

onSunday, 13 September, 2015 Permalink |[…] M is for measure, as in “measure theory” behind Lebesgue integration. There’s also a nice discussion of the paradoxical Cantor set that has dimension . […]

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## Reading the Comics, October 14, 2015: Shapes and Statistics Edition | nebusresearch 2:03 pm

onSaturday, 17 October, 2015 Permalink |[…] squares you could fit in a three-dimensional box. A two-dimensional object has no volume — zero measure, in three-dimensional space — so you could fit anything into it. This may be reasonable but it still runs against my […]

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## A Listing Of Mathematics Subjects I Have Covered In A To Z Sequences Of The Past | nebusresearch 4:01 pm

onThursday, 29 June, 2017 Permalink |[…] Measure […]

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