Sigma-algebras

Suppose that our pronoun model assigns a probability to the event that a form is plural. It must also assign a probability to the event that the form is not plural, since the second event is the complement of the first. If the model assigns probabilities to two events, it must also assign a probability to their union. The collection of measurable events must be closed under these operations.

A sigma-algebra, also called an event space, is a collection of events that contains the sample space and is closed under complements and countable unions. The model assigns probabilities only to events in this collection, which we write as \mathcal{F}.

To see why the collection needs structure, use the four outcome sample space

\Omega_4 \equiv\{\textit{he},\textit{him},\textit{they},\textit{them}\}.

Let P be the plural event:

P\equiv\{\textit{they},\textit{them}\}.

Its complement is

P^c=\{\textit{he},\textit{him}\}.

If plural number is the only distinction that the model measures, one possible sigma-algebra is

\mathcal{F}_P \equiv\{\varnothing,P,P^c,\Omega_4\}.

Plural and singular are measurable under \mathcal{F}_P. Accusative case is not, because the accusative event does not belong to \mathcal{F}_P.

Closure requirements

A collection \mathcal{F} is a sigma-algebra only if it satisfies three requirements.

The complete sample space \Omega belongs to \mathcal{F}.
If B belongs to \mathcal{F}, then its complement B^c also belongs to \mathcal{F}.
If B_1,B_2,\ldots belong to \mathcal{F}, then their union \bigcup_i B_i also belongs to \mathcal{F}.

The first requirement includes the certain event among the measurable events. The second includes the complement of every measurable event. The third includes the union of any countable collection of measurable events.

Some events follow from the three requirements. Since \Omega\in\mathcal{F}, closure under complements gives

\Omega^c=\varnothing\in\mathcal{F}.

Intersections follow from complements and unions. De Morgan’s law gives

B\cap C=(B^c\cup C^c)^c.

Thus a sigma-algebra that is closed under complements and countable unions is also closed under intersections.

A collection that is not a sigma-algebra

Consider the proposed collection

\mathcal{G}\equiv\{\Omega_4,P\}.

Check the proposed collection against each requirement.

It contains \Omega_4, so it passes the first requirement.
It contains P but not P^c, so it fails the second requirement.
It also lacks \varnothing, which must appear as \Omega_4^c.

Adding P^c and \varnothing repairs these failures and produces \mathcal{F}_P.

The following R list represents the four events in \mathcal{F}_P.

Code

omega <- c("he", "him", "they", "them")
plural <- c("they", "them")
singular <- setdiff(omega, plural)

event_space <- list(
  impossible = character(0),
  plural = plural,
  singular = singular,
  certain = omega
)

event_space

The list omits the singleton event \{\textit{them}\}. This set is a subset of \Omega_4, but it is not an event in \mathcal{F}_P. The model represents them as an outcome, but it cannot assign a probability to the event containing them alone.

Sample spaces and event spaces

\Omega states which individual outcomes exist in the model.
\mathcal{F} states which groupings of those outcomes can receive probabilities.

For a small finite sample space, we will often use the power set 2^\Omega, the collection containing every subset of \Omega. In that case, every possible grouping of the represented outcomes is measurable. The smaller sigma-algebra \mathcal{F}_P shows that this choice is not automatic.

An event outside the sigma-algebra

In the example above, the accusative event does not belong to \mathcal{F}_P. We thus cannot write its probability under a probability measure whose domain is \mathcal{F}_P. To assign a probability to accusative case, we must use a sigma-algebra that contains the accusative event.

For the finite examples in this course, taking \mathcal{F}\equiv2^\Omega makes every subset measurable. We still write \mathcal{F} separately because this choice is part of the probability-space specification.

Check your understanding

Start with \{\Omega_4,P\}. Which two events must be added to obtain \mathcal{F}_P?
Is \{\varnothing,\Omega_4\} a sigma-algebra? Check all three requirements.
Why does \{\textit{them}\}\subseteq\Omega_4 not guarantee that \{\textit{them}\}\in\mathcal{F}_P?
Use De Morgan’s law to explain why closure under complements and unions gives closure under intersections.

--- title: "Sigma-algebras" --- Suppose that our pronoun model assigns a probability to the event that a form is plural. It must also assign a probability to the event that the form is not plural, since the second event is the complement of the first. If the model assigns probabilities to two events, it must also assign a probability to their union. The collection of measurable events must be closed under these operations. A [**sigma-algebra**](https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_%28Siegrist%29/01%3A_Foundations/1.11%3A_Measurable_Spaces), also called an event space, is a collection of events that contains the sample space and is closed under complements and countable unions. The model assigns probabilities only to events in this collection, which we write as $\mathcal{F}$. To see why the collection needs structure, use the four outcome sample space $$ \Omega_4 \equiv\{\textit{he},\textit{him},\textit{they},\textit{them}\}. $$ Let $P$ be the plural event: $$ P\equiv\{\textit{they},\textit{them}\}. $$ Its complement is $$ P^c=\{\textit{he},\textit{him}\}. $$ If plural number is the only distinction that the model measures, one possible sigma-algebra is $$ \mathcal{F}_P \equiv\{\varnothing,P,P^c,\Omega_4\}. $$ Plural and singular are measurable under $\mathcal{F}_P$. Accusative case is not, because the accusative event does not belong to $\mathcal{F}_P$. ## Closure requirements A collection $\mathcal{F}$ is a sigma-algebra only if it satisfies three requirements. 1. The complete sample space $\Omega$ belongs to $\mathcal{F}$. 2. If $B$ belongs to $\mathcal{F}$, then its complement $B^c$ also belongs to $\mathcal{F}$. 3. If $B_1,B_2,\ldots$ belong to $\mathcal{F}$, then their union $\bigcup_i B_i$ also belongs to $\mathcal{F}$. The first requirement includes the certain event among the measurable events. The second includes the complement of every measurable event. The third includes the union of any countable collection of measurable events. Some events follow from the three requirements. Since $\Omega\in\mathcal{F}$, closure under complements gives $$ \Omega^c=\varnothing\in\mathcal{F}. $$ Intersections follow from complements and unions. De Morgan's law gives $$ B\cap C=(B^c\cup C^c)^c. $$ Thus a sigma-algebra that is closed under complements and countable unions is also closed under intersections. ## A collection that is not a sigma-algebra Consider the proposed collection $$ \mathcal{G}\equiv\{\Omega_4,P\}. $$ Check the proposed collection against each requirement. 1. It contains $\Omega_4$, so it passes the first requirement. 2. It contains $P$ but not $P^c$, so it fails the second requirement. 3. It also lacks $\varnothing$, which must appear as $\Omega_4^c$. Adding $P^c$ and $\varnothing$ repairs these failures and produces $\mathcal{F}_P$. The following R list represents the four events in $\mathcal{F}_P$. ```{r} #| eval: false omega <- c("he", "him", "they", "them") plural <- c("they", "them") singular <- setdiff(omega, plural) event_space <- list( impossible = character(0), plural = plural, singular = singular, certain = omega ) event_space ``` The list omits the singleton event $\{\textit{them}\}$. This set is a subset of $\Omega_4$, but it is not an event in $\mathcal{F}_P$. The model represents *them* as an outcome, but it cannot assign a probability to the event containing *them* alone. ## Sample spaces and event spaces 1. $\Omega$ states which individual outcomes exist in the model. 2. $\mathcal{F}$ states which groupings of those outcomes can receive probabilities. For a small finite sample space, we will often use the power set $2^\Omega$, the collection containing every subset of $\Omega$. In that case, every possible grouping of the represented outcomes is measurable. The smaller sigma-algebra $\mathcal{F}_P$ shows that this choice is not automatic. ## An event outside the sigma-algebra In the example above, the accusative event does not belong to $\mathcal{F}_P$. We thus cannot write its probability under a probability measure whose domain is $\mathcal{F}_P$. To assign a probability to accusative case, we must use a sigma-algebra that contains the accusative event. For the finite examples in this course, taking $\mathcal{F}\equiv2^\Omega$ makes every subset measurable. We still write $\mathcal{F}$ separately because this choice is part of the probability-space specification. ## Check your understanding 1. Start with $\{\Omega_4,P\}$. Which two events must be added to obtain $\mathcal{F}_P$? 2. Is $\{\varnothing,\Omega_4\}$ a sigma-algebra? Check all three requirements. 3. Why does $\{\textit{them}\}\subseteq\Omega_4$ not guarantee that $\{\textit{them}\}\in\mathcal{F}_P$? 4. Use De Morgan's law to explain why closure under complements and unions gives closure under intersections.