Constructing measures: overview – Serlo

We now know the basic concepts of measure theory and what is a measure on a $\sigma$-algebra. Our next big goal is to prove the existence and uniqueness theorems for measure continuations. This article summarizes the three most important questions we still have to answer and gives an overview of the procedure for the construction of measures.

Problem

Often one is in the situation to want to measure the volume of subsets of some basic set. The mathematical "measure" is then a map assigning a volume to as many subsets as possible. Depending in the case, it shall also fulfil some further properties.

Example (Measures with certain properties)

• We are looking for a measure $\mu$ that describes the probability of events occurring when, let's say, a fair die is rolled $n$ times. In addition, two properties should be satisfied: All one-element events should be equally probable, and the individual probabilities should sum to 1.
• Another example is the geometric length on $\mathbb {R}$: We are looking for a measure $\lambda$ which assigns a volume to as many subsets of $\mathbb {R}$ as possible. In addition, $\lambda$ should assign to every interval (i.e. $\lambda ([a,b])=b-a$) its length and be translation invariant (i.e. $\lambda (A)=\lambda (A+r)$ for all $r\in \mathbb {R}$).

• 
  A probability measure $\mu$assigns to sets of events $A,B$ their probability $0\leq \mu (A),\mu (B)\leq 1$. Here: rolling a dice twice.
• 
  Intervals of $\mathbb {R}$ shall get assigned their length $\lambda (A)$ by a suitable measure. This measure should be translation invariant on all sets.

We do not yet know whether a measure with the desired properties exists and on which (as large as possible) $\sigma$-algebra (set of subsets) it can be defined. To proceed, we have to deal with the following questions:

• Which $\sigma$-algebras are we allowed to choose as domain of definition? On the one hand, we want to be able to measure as many quantities as possible, so a large $\sigma$-algebra would be nice. On the other hand, we have already seen with the example of Banach-Tarski that a too large $\sigma$-algebra as domain of definition may destroy the existence of the measure.
• How can we ensure that the measure actually has the desired properties and at the same time is $\sigma$-additive? Are the desired properties at all compatible with $\sigma$-additivity, so a measure with these properties exists?
• Often $\sigma$-algebras are very large, even over-countable. It is not immediately clear how one can define the measure at all and thereby prescribe the values on the whole domain of definition. How can a mapping rule for the measure look like, in order to define it thereby uniquely?

Solution

Let's start with a "non-greedy" attitude and consider a smaller set system ${\mathcal {C}}\subseteq {\mathcal {P}}(\Omega )$ of subsets of $\Omega$. In particular, ${\mathcal {C}}$ need not yet be a $\sigma$-algebra. On this smaller set system we try to achieve the goal, i.e., to find a $\sigma$-additive function on sets on ${\mathcal {C}}$ that satisfies the desired properties. On the one hand, ${\mathcal {C}}$ must not be too large, so that we indeed find such a function on sets. On the other hand, ${\mathcal {C}}$ must be big enough to contain all the important information about the measure we are looking for. It seems reasonable to define ${\mathcal {C}}$ as a set system of "atomic" subsets (e.g., cuboids) which are easy to handle serve as building blocks for more complicated sets.

Example ("atomic" subsets)

• In the above example with the dice roll, ${\mathcal {C}}$ can be chosen, for example, as the set system of all one-element events.
• For the geometric length, ${\mathcal {C}}$ can be chosen as the set system containing all half-open intervals $(a,b]$ ($a,b\in \mathbb {R}$). Half-open intervals have the advantage over closed and open intervals that they can be joined without overlaps or gaps - a property that will prove helpful later in the actual construction of the measure.

On the other hand, it would not make sense to choose the set system ${\mathcal {C}}=\{\emptyset ,\mathbb {R} \}$ in the second example: It is true that we can easily find a measure with the desired properties. But ${\mathcal {C}}$ is not nearly does not contain information about the translation invariance and the property $\lambda ([a,b])=b-a$.

Then we continue the function on sets defined on ${\mathcal {C}}$ to obtain a measure on a $\sigma$-algebra. The question of the existence of a measure on a $\sigma$-algebra thus becomes the question of the existence of a continuation of a $\sigma$-additive function on sets from a smaller set system to a larger one. The following points remain to be investigated:

• To which $\sigma$-algebra can we continue, if we are starting from ${\mathcal {C}}$? We'll look at that in the article on generated $\sigma$-algebras.
• Does such a continuation beyond ${\mathcal {C}}$ even exist? We'll look at that in the article existence of a measure continuation.
• Is the continuation, and thus the measure we are looking for, already uniquely determined by the values on ${\mathcal {C}}$? We'll look at that in the article about the uniqueness of a measure continuation.