Measure Theoretic Probability - Video 6 - Four Examples
Four Examples
Length: measuring the length of subsets of real numbers (in a way that is consistent with the lengths of intervals).
Count: counting the number of elements in a subset of any set \(\Omega\).
Probability: For any sample space (probability space, whatever you want to call it) the measure of an event will be its probability.
Dirac: For any set \(\Omega\) (although usually a set that is geometric in interpretation, like the real numbers), and point \(x\in\Omega\), and for any mass (really, just any nonnegative real number) m, we define the Dirac measure
\(\delta_x(E)=m\) if \(x\in E\subseteq \Omega\) and
\(\delta_x(E)=0\) otherwise
Note that only length is a completely specific example. Count, probability, and Dirac are each classes of examples. There are many probability measures, for instance, you could have a probability measure for a coin toss, or a probability measure for a Gaussian process. There are many count measures, one for each choice of \(\Omega\) from which you would count. There are many Dirac measures, one for each choice of \(\Omega\) and then for each choice of \(x\in \Omega\).
The Universe
Any given measure must specify the universe in which it lives, so to speak. This is the set of all “points” in the universe. In the most general setting, this can be any set, which we often denote as \(\Omega\). In the setting of length measure, the universe is \(\Bbb R\). To express this we might write \(\Omega=\Bbb R\) to communicate that our universe, in this example, is the real numbers.
For these four canonical measures, there are sometimes special rules of nomenclature.
When the measure is a probability measure, we instead call the universe the sample space.
In the general case of any abstract measure, we will often refer to the elements of \(\Omega\) as points. That is to say x is a point if and only if \(x\in \Omega\). This is intended to conjure a mental image of a geometric space, but like with count measure, we should keep in mind that the space may not actually be geometric at all.
When the measure is a probability measure, we will instead call the points atomic events.
Measurable subsets
We have already seen that, in the example of length measure, we will not assign to every subset a measure. That is to say, not everything in \(\mathcal P(\Bbb R\)\) will be measurable. There will instead exist a distinguished sub-collection of measurable sets, and the others we will not attempt to assign measure. (The precise definition of this subset is to come much later in this series.)
Therefore in the more general setting of any measure, we will always specify a collection of measurable subsets. For length measure we will give its measurable subsets the symbol \(\mathcal L\). Therefore \(\mathcal L\subseteq \mathcal P(\Bbb R)\) will be the distinguished collection of measurable subsets.
In the more general setting of an abstract measure space, the measurable subsets will be denoted by \(\mathcal M\) and will always be \(\mathcal M\subseteq\mathcal P(\Omega)\). However, not just any such sub-collection of subsets will be permitted. The precise requires will come up in a video soon. We call \(\mathcal M\) the collection of measurable subsets or more briefly the measurables. Any element of \(\mathcal M\) is called a measurable subset or more briefly a measurable.
For probability measures, we will instead call the measurables the events, and any element of the events is called an event.
For counting and Dirac measure we will always take \(\mathcal M=\mathcal P(\Omega)\). That is to say, for these examples (and some others) every subset of the universe is a measurable.
The Measure Function
Finally, the object that actually measures any given measurable is called the measure function or more briefly the measure. It always has domain the measurables. In the general setting, we will denote it by \(\mu\),
\[ \mu : \mathcal M\to \Bbb R^*\]
Note that \(\Bbb R^*\) is the extended real numbers \(\Bbb R^* = \Bbb R\cup \{-\infty,\infty\} \). This is to accommodate some (in fact, many) situations in which a set has infinite measure. For instance with length measure, the measure of the entire real line should be \(\infty\) since it is infinite in length.
For length measure, we will denote its measure function by m. So
\[m:\mathcal L\to \Bbb R^*\]
For count measure, we will denote its measure function by c. For probability measure, we will denote its measure function by p. For Dirac measure (at a point x) we will denote its measure function by \(\delta_x\).
We will also say that the measure of the set \(E\in\mathcal M\) is the value \(\mu(E)\). (So for instance, the length measure of the interval (-2,2) is 4. The count-measure of (-2,2) is \(\infty\) and the count measure of {1,2} is 2.)