# What is volume, exactly?

# Introduction

Can you define the notion of a volume of a shape? For example, what does ”volume” mean for a ball of radius 3? If you think about it, this is not a simple question. You probably understand intuitively what ”volume” and ”area” mean, but giving a proper definition might be tricky. The goal of this article is to provide you a view of a more formal mindset about defining volume and area, and to give you a brief introduction to measure theory.

# Back to school

To begin with, let’s go way back to when you first learned about volume. In fact, let’s concentrate on area instead of volume right now for simplicity, since volume is essentially a higher-dimensional extension of area.

When you first learn about area in your Maths class, your teacher probably gives you this definition:

Areaof a shape is the number of 1x1 squares needed to completely tile the shape.

And then they probably draw a picture like this:

This definition works perfectly fine (on the intuitive level, of course) until you get to triangles, rectangles with non-whole sides, or other weird polygons:

For example, you can’t tile this rectangle completely with whole 1x1 squares, so, according to our current definition, its area is undefined. Here, the teacher says that breaking the squares into pieces is allowed, so under the updated definition, this rectangle now has a defined area.

Additionally, the teacher normally also says that area possesses a property of **additivity**, that is, if you take two non-intersecting shapes and ”glue” them together into a single big shape, its area will be equal to the sum of the areas of these shapes.

But why though? Intuitively, this makes sense, but it is extremely hard, if not impossible to derive from our definition, since both definitions rely heavily on intuition. If we assume additivity, however, we can prove a lot of useful things, for example how rearranging pieces or breaking up the 1x1 squares differently does not change the area of a shape.

But the worst part begins when you start learning about circles, since now you have to break 1x1 squares infinitely many times in order to calculate an area, and infinity **is not an intuitive concept at all**, so now you’re trying to apply intuition to something that is inherently non-intuitive.

In this case, because of additivity, an area of a circle can be represented as an infinite sum. This, however, creates a lot of questions, that are left unanswered. Why does this sum always converge? That is, why can we assign a value to this sum? What if, as we add more terms, the sum oscillates around some value, or even tends to infinity? Why does it converge to exactly 𝜋𝑟²? (Admittedly, some teachers *do* try to explain this one, and I genuinely applaud them for it)

Defining and calculating areas of more complex shapes is beyond the school curriculum.

To summarize, in school we intuitively defined an area of a shape by tiling it with 1x1 squares or pieces of them. This definition left a lot of questions, which are impossible to answer, due to the ambiguity of the definition. The definition of a volume of a shape is similar, we just have to tile with multidimensional cubes with sides of length 1.

# Starting to formalize our definition

Let’s focus on only defining volume, and saying that area is just a two-dimensional variant of volume, for the sake of simplicity.

Firstly, let’s formalize the notion of a shape. We will define volume for subsets of ℝ*ⁿ* (set of tuples of length 𝑛, where each element is a real number, for example, (1, 𝜋, 3.21) ∈ ℝ³). This way, we can think of an 𝑛-dimensional shape as a collection of ”points”, represented as tuples of length 𝑛, in ℝ*ⁿ*.

Before we can proceed with the formal definition of volume, let’s consider the properties that we ”want” it to have, that is, which intuitive behaviours do we want to formalize.

There are only two intuitive properties, that we will consider:

- Volume is non-negative, that is, for any subset 𝐴 of ℝ
*ⁿ*(so, for any set of ”points”), the volume of 𝐴 is greater than or equal to 0. - Volume is additive, that is, for two non-intersecting sets 𝐴 and 𝐵, such that both of them are subsets of ℝ
*ⁿ*(so, for two non-intersecting sets of ”points”), the volume of their union 𝐴 ∪ 𝐵 is equal to the sum of the volume of 𝐴 and the volume of 𝐵.

If a function 𝜇 that takes a subset of ℝ*ⁿ* as an argument satisfies both of these properties, let’s call it a **measure **on ℝ*ⁿ *(note that we can define measures on other sets, too, but the definition will be slightly different). Now, we need to find a measure on ℝ*ⁿ* that models our informal definition of volume, namely tiling with 𝑛-dimensional cubes or their pieces.

# Rectangles and their Jordan measure

Let’s consider real-valued intervals of type [𝑎, 𝑏) = {all real 𝑥, such that 𝑎 ≤ 𝑥 < 𝑏} (it is important to note that we only consider intervals, where 𝑎 ≤ 𝑏, so let’s invalidate all other notation), more specifically, Cartesian products of these intervals, that is,

(Notice how ℝ*ⁿ* from before is just ℝ × . . . × ℝ repeated 𝑛 times)

To visualize, a Cartesian product of these intervals is a multidimensional rectangle but without a few of its sides, since we ”leave out” the right bounds in the intervals. In the two-dimensional case, [𝑎, 𝑏) × [𝑐, 𝑑) can be visualized like this on the 𝑋𝑌 plane:

So, from now on, let’s call Cartesian products of such intervals **rectangles**.

Consider a rectangle 𝑅 = [𝑎₁, 𝑏₁) × [𝑎₂, 𝑏₂) × . . . × [𝑎*ₙ*, 𝑏*ₙ*). Let’s define its **Jordan measure** like so: 𝜇(𝑅) = (𝑏₁ − 𝑎₁)(𝑏₂ − 𝑎₂) . . . (𝑏*ₙ* − 𝑎*ₙ*). As you can see, its Jordan measure is exactly the same as our intuitive understanding of volume would suggest, just the product of the lengths of its sides. Notice

that 𝜇(𝑅) is non-negative, since 𝑎*ᵢ* ≤ 𝑏*ᵢ* for all 𝑖 = 1, 2, 3, . . . , 𝑛.

Let’s now define a** Jordan measure** of a union of disjoint rectangles like so: if 𝑅₁,𝑅₂, . . .,𝑅*ₙ* are disjoint rectangles, then 𝜇(𝑅₁ ∪ 𝑅₂ ∪ . . .∪𝑅*ₙ*) = 𝜇(𝑅₁) + 𝜇(𝑅₂) + . . . + 𝜇(𝑅*ₙ*). Now we’ve covered both properties of a measure (non-negativity and additivity), so the Jordan measure of rectangles is indeed a measure, by definition.

Let’s consider an example:

Here, we have two rectangles: 𝑅₁ = [𝑎, 𝑏) × [𝑎′, 𝑏′) with Jordan measure 𝜇₁ and 𝑅₂ = [𝑐, 𝑑) × [𝑐′, 𝑑′) with Jordan measure 𝜇₂.

Firstly, according to our definition of a Jordan measure of a rectangle,

𝜇₁ = (𝑏 − 𝑎)(𝑏′ − 𝑎′) and 𝜇₂ = (𝑑 − 𝑐)(𝑑′ − 𝑐′). Secondly, according to our definition of a Jordan measure of a union of disjoint rectangles,

𝜇(𝑅₁ ∪ 𝑅₂) = 𝜇₁ + 𝜇₂ = (𝑏 − 𝑎)(𝑏′ − 𝑎′) + (𝑑 − 𝑐)(𝑑′ − 𝑐′).

Additionally, notice that if 𝑅₁ is a 1x1 square, that is, if 𝑏 = 𝑎 + 1 and

𝑏′ = 𝑎′ + 1, then 𝜇(𝑅₁) = 1, which corresponds to our intuitive definition of volume from before. In fact, because of this and because of the additivity property of the Jordan measure, we’ve managed to formalize ”tiling” from the intuitive definition of volume for rectangles!

# Bounded sets, infimum, and supremum

Before we can proceed to defining the Jordan measure on arbitrary sets, we need to understand what a bounded set is, and also what ”infimum” and ”supremum” are.** If you are already familiar with these topics, you can safely skip this part.**

Firstly, let’s discuss bounded sets. We obviously only want to define volume for finite objects, and not for something like an infinitely tall cylinder. That is why we will only define the general Jordan measure on so-called bounded sets. A set 𝑆 (in multidimensional space of dimensionality n, in our case ℝ*ⁿ*) is called a **bounded set**, if there exists a ”maximum” distance between any two ”points” in this set, that is, there exists a real number 𝑑, such that for any two ”points” 𝑎, 𝑏 ∈ 𝑆 the distance between them is less than 𝑑 (assuming that ”distance” is defined for S, for ℝ*ⁿ* we have the usual Euclidean distance).

Secondly, let’s discuss infimums and supremums. Let’s assume that order ”≤” is defined for a set 𝑆 (for example, this is true for 𝑆 = ℝ). Then, an **infimum **of a subset 𝑠 of set 𝑆 is the greatest element in 𝑆, such that it is less than or equal to every element of 𝑠. Essentially, infimum is the generalization

of the notion of a minimum, since a subset might not have a minimum element, but might have an infimum. For example,

inf{𝑥 ∈ ℝ, such that 0 < 𝑥 < 1} = 0, but this set does not have a minimum

element. If a set *does *have a minimum element, then its infimum is equal to the minimum element.

**Supremum **is defined similarly, it is a generalization of the notion of a maximum. For example, sup{𝑥 ∈ R, such that 0 < 𝑥 < 1} = 1. If a set has a maximum element, then its supremum is equal to the maximum element.

# Jordan measure of abstract sets

Since we already formally defined volume for rectangles, let’s use them to ”approximate” the volume of all other shapes (and sets, for which it ”makes sense” to have volume defined for).

Notice that if we have a (finite) union of arbitrary rectangles (not necessarily disjoint), we can always subdivide these rectangles and join the parts in such a way, that we end up with a union of disjoint rectangles:

Now, for a bounded set 𝑆 let’s define its

**Outer Jordan measure:**

**Inner Jordan measure:**

Roughly speaking, the outer Jordan measure of a bounded set 𝑆 is the ”outer” approximation of the volume of 𝑆, since we consider only rectangle unions that contain 𝑆, and the inner Jordan measure of 𝑆 is the ”inner” approximation of the volume of 𝑆, since we consider only rectangle unions that are contained inside of 𝑆.

A bounded set 𝑆 is **Jordan measurable**, if

So 𝑆 is Jordan measurable, if those ”approximations” are equal to each other. If 𝑆 is Jordan measurable, we define its **Jordan measure** as just

So the Jordan measure of 𝑆 is defined as either of the ”approximations” (doesn’t matter which, since they are equal).

If you think about it, this method of defining volume formally for all shapes is more or less what we did in school: we know that the Jordan measure of a 1x1x. . .x1 cube is 1, so we can work with rectangles now. Instead of breaking the unit cube into arbitrary pieces, we just consider rectangles with non-whole sides, and try to ”tile” the shape with them. That’s how we get the inner Jordan measure, and since all ”intuitive” shapes are Jordan measurable, the ”volume” of the shape, that is, its Jordan measure, is just equal to its inner Jordan measure. Therefore, we’ve formally defined volume (and, consequently, area) for all ”intuitive” shapes.

# Beyond Jordan measure

But are there any sets that are not Jordan measurable? Turns out, there are. For example, the set 𝑆 = {all rational 𝑥, such that 𝑥 ∈ [0, 1]} is not Jordan measurable.

Since 0 and 1 are rational numbers, the outer Jordan measure of 𝑆 is equal to 1, since we have to consider the rectangles of type [0, 1 + 𝑥), where 𝑥 > 0 (since both 0 and 1 have to be included), and the infimum of Jordan measures of such rectangles is 1.

Since between every two real numbers there exists a rational number and an irrational number, 𝑆 is just a set of infinitely many separated points (that is, no interval or segment is a subset of 𝑆). Since in the definition of the Jordan measure we stated that we only consider **finite **unions of rectangles, there

is only one such union that is contained in 𝑆 — the empty set. Since the empty set can be represented as, for example [1, 1), its Jordan measure is 0, so the inner Jordan measure of 𝑆 is equal to 0.

The inner and the outer Jordan measures of 𝑆 are not equal, therefore, by definition, 𝑆 is not Jordan measurable.

If we extend our view to **countable **unions of rectangles (that is, infinite unions of the same size as the set of natural numbers), we get a **Lebesgue measure**. Every Jordan measurable set is also Lebesgue measurable, and both measures for such set are equal, so the Lebesgue measure allows us to work with a much larger class of sets, compared to the Jordan measure. The set 𝑆 turns out to be Lebesgue measurable, even though it is not Jordan measurable.

# Conclusion

To summarize, we considered a small number of intuitive properties that a ”correct” volume function should have, namely, volume is non-negative, volume is additive, the volume of a 1x1x. . .x1 cube is equal to 1, and the volume of a rectangle is equal to the product of the lengths of its sides. Then we crafted a formal definition of volume that satisfies these intuitive properties, which we also formalized.

This formal definition allows us to answer all of the questions about volume that the intuitive school definition is simply unable to answer. Additionally, now we can use volume as a valid mathematical tool in our theorems or proofs without being afraid of ambiguities that an informal definition might have.