Overview: Objects in Measure Theory – "Math for Non-Geeks"

In the following articles on measure theory, we will gradually introduce various mathematical objects. These articles tell a story in which we will follow possible considerations of a mathematician, so we introduce new objects only when we really need them.

This article summarizes the objects in a compact way, so you can easily compare them.

Set systems

The basis of measure theory is always a large basic set $\mathrm{\Omega }$, for which we want to assign a measure to subsets $A\subseteq \mathrm{\Omega }$ that is, a number $\mu (A)$ that indicates how large $A$ is. In many cases, however, not every subset $A$ is suitable for such an assignment. For instance the Banach-Tarski paradox or the Vitali sets show this.

Those sets to which, we can assign a suitable measure $\mu (A)$ will simply be called measurable. We put them into a set system $𝒜$. So the $𝒜$ is a set containing sets itself (like a bag in which there are even more bags), e.g. $𝒜=A_1,A_2,A_3,\mathrm{...}$ with $A_1,A_2,A_3\subseteq \mathrm{\Omega }$.

To do computations with measures (i.e., addition, subtraction), we would like to perform operations with the sets, like unions $A_1\cup A_2$, intersections $A_1\cap A_2$ or taking complements $A^{\complement }$. And this without "getting kicked out of $𝒜$", if possible. In mathematics, one therefore classifies set systems $𝒜$ into different types, depending on how many operations we can perform without getting kicked out of $𝒜$ and on other nice properties which they may satisfy:

Types of set systems. An arrow means "is a generalization of".

• The $\sigma$-algebra is the most special and most frequently encountered set system type. Here we can "afford relatively many operations" which avoids problems of the kind "$\mu (A)$ is not defined on this set". $𝒜$ is an$\sigma$-algebra if and only if

1. $\mathrm{\Omega }\in 𝒜$
2. $A,B\in 𝒜⟹A\setminus B\in 𝒜$
3. If a sequence of sets $(A_n{)}_{n\in \mathbb{N}}$ is in $𝒜$ , then also $\bigcup _{n\in \mathbb{N}}{A}_n\in 𝒜$

• An algebra (of sets) also satisfies these 3 axioms, but the 3rd is only required to hold for finite sequences $(A_n{)}_{n\in \{1,\mathrm{...},N\}}$. The "$\sigma$" here stands for a countably infinite union of sets. If one omits it, then "only" finite sequences are allowed and one gets a more general set system. That is, there are "more algebras than $\sigma$-algebras". A set system $𝒜$ is an algebra if and only if

1. $\mathrm{\Omega }\in 𝒜$
2. $A,B\in 𝒜⟹A\setminus B\in 𝒜$
3. If a sequence of sets $(A_n{)}_{n\in \{1,\dots ,N\}}$ is in $𝒜$ , then also $\bigcup _{n=1}^N{A}_n\in 𝒜$

• A $\sigma$-ring (also denoted $ℛ$) satisfies all conditions of the $\sigma$-algebra, except for 1. That is, we also allow set systems containing only smaller sets. E.g., there could be a maximal set ${\mathrm{\Omega }}_R\subset \mathrm{\Omega }$ containing all $A\in ℛ$. A system $ℛ$ is a $\sigma$ ring if and only if

1. $A,B\in ℛ⟹A\setminus B\in ℛ$
2. Whenever a sequence of sets $(A_n{)}_{n\in \mathbb{N}}$ is in $ℛ$ , then we also have $\bigcup _{n\in \mathbb{N}}{A}_n\in ℛ$

Sometimes it is required as an additional condition that $ℛ$ is not empty, i.e. $ℛ\ne \mathrm{\varnothing }$. As soon as a ring $ℛ$ contains any set $A$, it always contains also the empty set $\mathrm{\varnothing }=A\setminus A$.

• A ring (of sets) $ℛ$ is obtained equivalently by taking away from the $\sigma$-ring the $\sigma$-property. That means, we allow only finite $(A_n{)}_{n\in \{1,\mathrm{...},N\}}$ in our condition. From the axioms of the $\sigma$-algebra only the 2nd and the 3rd in "finite" form are valid:

1. $A,B\in ℛ⟹A\setminus B\in ℛ$
2. If a sequence of sets $(A_n{)}_{n\in \{1,\dots ,N\}}$ is in $ℛ$ , then also $\bigcup _{n=1}^N{A}_n\in ℛ$

• * The Dynkin-system $𝒟$ is a separate type of set system. We will need it later to describe when measures match. The 3 axioms are:

1. $\mathrm{\Omega }\in 𝒟$
2. For every pair of sets $A,B\in 𝒟$ with $A\subseteq B$ we have $B\setminus A\in 𝒟$
3. For countably many, pairwise disjoint sets $A_1,A_2,\dots \in 𝒟$ we have $\underset{n\in \mathbb{N}}{⨄}{A}_n\in 𝒟$.

Further set systems, that do not appear in the articles, are:

• The monotone class $ℳ$: a type of set system containing all limits of monotonically increasing or decreasing set sequences $(A_n{)}_{n\in \mathbb{N}}$. That means,

1. If $A_1,A_2,\dots \in ℳ$ form a monotonically increasing sequence, i.e., $A_1\subseteq A_2\subseteq \mathrm{...}$, then we have $\bigcup _{n\in \mathbb{N}}{A}_n\in ℳ$
2. If $A_1,A_2,\dots \in ℳ$ form a monotonically decreasing sequence, i.e., $A_1\supseteq A_2\supseteq \mathrm{...}$, then we have $\bigcup _{n\in \mathbb{N}}{A}_n\in ℳ$

• The semi-ring is a generalization of the ring. The essential point is that $A\setminus B$ no longer lies in $ℛ$, but only needs to be represented by a disjoint union of sets from it. The condition $\mathrm{\varnothing }\in ℛ$ (which always holds on rings) must thus be required separately. Instead of union stability one also demands cut stability:

1. $\mathrm{\varnothing }\in ℛ$
2. $A,B\in ℛ⟹A\cap B\in ℛ$
3. For $A,B\in ℛ$ there exist disjoint sets $C_1,\dots ,C_n\in ℛ$, such that $A\setminus B=\bigcup _{n=1}^N{C}_n\in ℛ$

Functions on sets

Additive functions on sets

On the set systems defined above, we now try to define functions $\mu$ (or $\eta$) that intuitively measure the "size" of a set. The intuition "measure a size" can be translated into several desirable properties. For example, an empty set should have size 0, so $\mu (\mathrm{\varnothing })=0$. The more of these properties hold, the more the function $\mu$ matches our intuition to measure the size of sets.

Depending on how many and which of these desirable properties are satisfied by $\mu$, we divide these functions into different classes. The most specific class is the measure, which has relatively many good properties and is therefore often used in mathematics, e.g., with in probability theory and statistics.

Relationship between 4 set systems. An arrow means "is a generalization of".

• A measure $\mu$ is a function on a $\sigma$-algebra $𝒜$ with the quite intuitive property that the empty set has measure 0 and when joining sets that do not overlap, their measures must also be added:

1. $\mu (\mathrm{\varnothing })=0$
2. $\mu$ is $\sigma$-additive, i.e., $\mu \left(\underset{n\in \mathbb{N}}{⨄}{A}_n\right)=\sum _{n\in \mathbb{N}}\mu (A_n)$

• A pre-measure $\mu$ is in principle the same as a measure, but needs to be defined only on a $\sigma$-ring $ℛ$ . So the set $\mathrm{\Omega }$ can be in $ℛ$, but doesn't need to be. Thus it holds that

1. $\mu (\mathrm{\varnothing })=0$
2. $\mu$ is $\sigma$-additive, i.e., $\mu \left(\underset{n\in \mathbb{N}}{⨄}{A}_n\right)=\sum _{n\in \mathbb{N}}\mu (A_n)$

• A volume $\mu$ on a ring $ℛ$ is a kind of "measure without $\sigma$-property for additivity". So we require the additivity only for finite unions:

1. $\mu (\mathrm{\varnothing })=0$
2. $\mu$ is additive, i.e., $\mu \left(\underset{n=1}{\overset{N}{⨄}}{A}_n\right)=\sum _{n=1}^N\mu (A_n)$

• A continuous volume $\mu$ on a ring $ℛ$ needs - as the name suggests - to be continuous:

1. $\mu (\mathrm{\varnothing })=0$
2. $\mu$ is additive
3. $\underset{n\to \mathrm{\infty }}{lim}\mu (A_n)=A$, as soon as a sequence of sets $(A_n{)}_{n\in \mathbb{N}}$ converges monotonically increasingly or decreasingly to $A$ . For monotonically decreasing sequences, we further require $\mu (A_n)<\mathrm{\infty }$ nötig.

In a sense a measure is a "$\sigma$-volume". However, since measures appear much more often in mathematics than volumes, they are given an own name.

Subadditive functions on sets

The following two classes of functions are not additive, but only sub-additive and therefore get their own letter $\eta$. I.e., if one unites, for example, a set with $\eta (A)=1$ and one with $\eta (B)=2$ there could be $\eta (A\cup B)=4$ (or we could have any number $\ge 3$)! This contradicts our intuition of "measuring a size". Therefore, we give the functions a separate symbol $\eta$.

Every outer measure is an outer volume

• The outer volume on the power set $𝒫(\mathrm{\Omega })$ is defined in analogy to the volume above. But instead of additivity, one only requires sub-additivity:

1. $\eta (\mathrm{\varnothing })=0$
2. $\eta$ is sub-additive, so from $A\subseteq \bigcup _{n=1}^N{A}_n$ we get $\eta (A)\le \sum _{n=1}^N\eta (A_n)$

• An outer measure on the power set $𝒫(\mathrm{\Omega })$ is the $\sigma$-version of the outer volume. That is, subadditivity is required even for countably infinite unions instead of finite unions. Thus, the sub-additivity becomes a $\sigma$-sub-additivity, where the $\sigma$ stands for "countably infinitely many".

1. $\eta (\mathrm{\varnothing })=0$
2. $\eta$ is $\sigma$-sub-additive, so from $A\subseteq \bigcup _{n=1}^{\mathrm{\infty }}{A}_n$ we get $\eta (A)\le \sum _{n=1}^{\mathrm{\infty }}\eta (A_n)$

Examples: separating set system definitions

The definitions of the set systems above are quite abstract, and it is not obvious why some of them might not be equivalent. In the following examples, we will see how to separate the set system types

$\text{ring}\leftrightarrow \sigma \text{-ring}\leftrightarrow \text{algebra}\leftrightarrow \sigma \text{-algebra}$

That means, we find examples that are of one but not a second type, respectively. In addition, you may find some (out of many possible) visualizations for those abstract definitions.

Example (Rings vs. $\sigma$-rings)

We construct a set system in which "limiting sets" are not contained: Let $(a_n{)}_{n\in \mathbb{N}}=1-2^{-n+1}$ be a sequence of real numbers. These are elements of the interval $[0,1)$. We choose the basic set $\mathrm{\Omega }:=[0,2)$ to be slightly larger (this is perfectly allowed) and define the set system

${ℛ}_0=\{[a_j,a_{j+1}),j\in \mathbb{N}\},$

i.e., all half-open intervals with endpoints from the sequence. This is not yet a ring. But we can make it a ring:

Let $ℛ$ be the set system of all finite unions and intersections of intervals from ${ℛ}_0$ (which are again finite unions of half-open intervals with $a_j$ as endpoints). The system $ℛ$ is a ring (the "ring generated by ${ℛ}_0$"), but $ℛ$ is not a $\sigma$-ring: the limiting sets

$[a_j,1)=\underset{k\to \mathrm{\infty }}{lim}[a_j,a_k),j\in \mathbb{N}$

are not included. However, we can turn $ℛ$ into a $\sigma$-ring by including the "limiting sets" $[a_j,1)=\underset{k\to \mathrm{\infty }}{lim}[a_j,a_k)$ for all $j\in \mathbb{N}$. The set system

${ℛ}_{\sigma }=ℛ\cup \{[a_j,1)=\underset{k\to \mathrm{\infty }}{lim}[a_j,a_k),j\in \mathbb{N}\}$

is again a $\sigma$-ring (and thus also a ring).

• 
  The system $ℛ$ is "only" a ring (of sets).
• 
  The system ${ℛ}_{\sigma }$ is at the same time a $\sigma$-ring.

Example ($\sigma$-algebras vs. $\sigma$-rings)

We copy the setting from the previous example: The set system ${ℛ}_0$ consists of all intervals $[a_j,a_{j+1})$ with endpoints in the sequence $(a_n{)}_{n\in \mathbb{N}}=1-2^{-n+1}$ . We choose the basis set to be $\mathrm{\Omega }:=[0,2)$ (see figure).

• 
  The set system ${𝒜}_{\sigma ,2}$ is a $\sigma$-algebra.
• 
  The set system ${ℛ}_{\sigma }$ is "only" a $\sigma$-ring.

The set system ${ℛ}_0$ generates a ring $ℛ$ (by finite unions and intersections). Joining it with the limit sets $[a_j,1)=\underset{k\to \mathrm{\infty }}{lim}[a_j,a_k)$ yields a $\sigma$-ring

${ℛ}_{\sigma }=ℛ\cup \{[a_j,1)=\underset{k\to \mathrm{\infty }}{lim}[a_j,a_k),j\in \mathbb{N}\}$

However, the set system ${ℛ}_{\sigma }$ is not a $\sigma$-algebra, since it does not contain the basic set $\mathrm{\Omega }$ (likewise the empty set $\mathrm{\varnothing }$ is not contained). In a sense, the set system is "too small". We can make it a $\sigma$-algebra: For this we add $\mathrm{\Omega }$ and the empty set $\mathrm{\varnothing }$, as well as the complement sets $R^{\complement }:=\mathrm{\Omega }\setminus R$ for all $R\in {ℛ}_{\sigma }$. From these sets we again form all arbitrary unions us intersections, and add them to the set system.

The result is a $\sigma$-algebra ${𝒜}_{\sigma }$ consisting of arbitrary unions of half-open intervals, with endpoints in $\{2,1,a_1,a_2,a_3\mathrm{....}\}$.

The only difference to ${ℛ}_{\sigma }$ is that 2 is also allowed as endpoint and the empty set $\mathrm{\varnothing }$ is included. $\sigma$-algebras can hence thought to be larger than mere $\sigma$-rings.

We can make ${𝒜}_{\sigma }$ even larger by adding a finite set of arbitrary points $\{b_1,\mathrm{...},b_n\}$ to the set of allowed endpoints. From this we can construct a "larger $\sigma$ algebra" ${𝒜}_{\sigma ,2}$ by choosing ${𝒜}_{\sigma ,2}$ as a set system of arbitrary unions of half-open intervals between the endpoints $\{2,1,b_1,\mathrm{...},b_n,a_1,a_2,a_3\mathrm{...}\}$ (which is shown in the figure above).

Example ($\sigma$-algebras vs. algebras)

We use again the setting of the two examples above: On the interval $\mathrm{\Omega }=[0,2)$ we define a sequence $(a_n{)}_{n\in \mathbb{N}}$ of endpoints $a_n=1-2^{-n+1}$.

The set system $ℛ$, which consists of arbitrary finite unions of intervals of the form $[a_j,a_k),j<k$ is a ring of sets. However, it is not a $\sigma$ ring.

If we add 1 to the possible endpoints $a_1,a_2,\dots$ (that is, $1,a_1,a_2,\dots$ are allowed as endpoints of half-open intervals), and also allow countably infinite unions, we obtain a $\sigma$-ring ${ℛ}_{\sigma }$. However, it is not a $\sigma$-algebra, since it does not contain $\mathrm{\Omega }$.

If we add another 2 to the possible endpoints, we get a $\sigma$-algebra ${𝒜}_{\sigma ,2}$. Now, if we add finitely many more arbitrary endpoints, we still end up with $\sigma$-algebras, which are, however, larger, and will be denoted by ${𝒜}_{\sigma ,2}$.

The $\sigma$-algebras ${𝒜}_{\sigma }$ and ${𝒜}_{\sigma ,2}$ can be very easily transformed back into an algebra by taking out the endpoint 1.

We define the set system $𝒜$ to contain any countable unions of intervals with the same endpoints as those from ${𝒜}_{\sigma }$, except 1. Then $𝒜$ is an algebra, because $𝒜$ contains $\mathrm{\Omega }$. But taking it out destroys the $\sigma$-property. That is, a countable union of sets (which contains, say, 1 as an endpoint) need no longer be in $𝒜$.

Similarly, we can transform the $\sigma$-algebra ${𝒜}_{\sigma ,2}$ into an algebra ${𝒜}_{\sigma ,2}$ by removing the endpoint 1. The $\sigma$-property will then no longer hold.

• 
  The system ${𝒜}_{\sigma ,2}$ is a $\sigma$-algebra.
• 
  If the endpoint 1 is taken away, "only an algebra" remains.