Comments on Can expectation be calculated if the events aren't numbered?
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Can expectation be calculated if the events aren't numbered?
Assuming I have a dice whose sides are named, not numbered. Can I have some way to describe the expectation of it? Or must I numbered the events in order to get the mean?
To put it in another way: why must the codomain of the measure be $\mathbb R$ (or $\mathbb R \cup \{∞\}$ to be precise), beside practical applications? What properties do we lose or gain if it is not $\mathbb R$? What if the measure is not a Lebesgue measure as well?
Note that the probability to get each side is still $\frac{1}{6}\in \mathbb R$.
Related question: How important results in classical statistics be generalized when codomain of the measure is not $\mathbb R$?
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| User | Comment | Date |
|---|---|---|
| Ooker | (no comment) | May 5, 2026 at 06:56 |
Given that the expectation is simply an integral (or, rather, "simply"...), you can certainly generalize it to any circumstance where integration makes sense. For example, suppose you roll two dice, and take the result as a random vector $(x(\omega), y(\omega))$. Then we can ask about the expected value of the result and calculate that as per Lebesgue integration.
If you are looking for the most general possible setting where you could do this, then certainly you need a notion of addition on the codomain, and defining what a random variable is would require some notion of measurability of function. Also, if we want to talk about the convergence of expectations, or any other sense of their distance, we would want the codomain to be at least a topological or metric space.
This quickly leads to having a metric measure space as a requirement for building up a theory, and to get sequences to converge, we would want it to be complete, and soon enough we are mostly talking about Banach spaces.
Considering the real line does have benefits. For example, the notion of a cumulative distribution function relies on the target being an ordered space. Also, we are interested in probabilities, which take values on the real interval $[0, 1]$, so we will often end up mapping stuff to the real line regardless.
As a concrete example, Brownian motion is typically defined as a mapping from a filtered probability space (with time playing a role) to some Euclidian space, but also a Riemannian manifold could be used. To study the properties of Brownian motion, for example, whether it keeps returning to a small neighbourhood of the starting point, we use the metric structure of the manifold. We might want to calculate the expected values of its distance from the starting point as a function of time, and here we end up a composing the Brownian motion with a distance from a point, which then takes values on the real line. So the composition is a real-valued random variable.
Lebesgue measure
Probability theory typically works with Borel sigma-algebrae. Borel and Lebesgue measures agree where both are defined, but not all null sets are Borel-measurable. So, in a sense, they are the same measure up to null sets. This difference is not important at first when one studies stochastics, but later various ways of extending measures become important, as the stochastic processes get complicated and the underlying probability spaces can become infinite-dimensional.
I do not recall the details of this anymore.
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