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The Borel sigma-algebra on an interval $I \subseteq \mathbb{R}$ is defined as the smallest sigma-algebra that contains all open intervals $(a,b)$ contained in $I$. It is denoted as $\mathcal{B}(I)$.

Conditions: $I$ is an interval on the real line $\mathbb{R}$.; The generating sets are open intervals $(a,b) \subseteq I$.

The Borel sigma-algebra on an interval $I \subseteq \mathbb{R}$ is defined as the smallest sigma-algebra that contains all open intervals $(a,b)$ contained in $I$. It is denoted as $\mathcal{B}(I)$.

Conditions: $I$ is an interval on the real line $\mathbb{R}$.; The generating sets are open intervals $(a,b) \subseteq I$.

When the sample space $\Omega$ is finite or countable, a suitable sigma-algebra should typically contain all singletons, meaning sets of the form $\{x\}$ for every $x \in \Omega$. This allows probabilities to be assigned to all individual outcomes.

Conditions: $\Omega$ is finite or countable.; The goal is to assign probabilities to individual outcomes.

When the sample space $\Omega$ is finite or countable, a suitable sigma-algebra should typically contain all singletons, meaning sets of the form $\{x\}$ for every $x \in \Omega$. This allows probabilities to be assigned to all individual outcomes.

Conditions: $\Omega$ is finite or countable.; The goal is to assign probabilities to individual outcomes.

The Borel sigma-algebra is the standard choice in continuous probability because there exists a unique probability measure on $\mathcal{B}([0,1])$ that assigns to each half-open interval $(a,b]$ its exact length $b-a$. This measure, known as Lebesgue measure, corresponds to the uniform distribution and avoids the non-measurability issues of the power set.

Conditions: The context is the continuous setting.; The measure is defined on $\mathcal{B}([0,1])$.; The endpoint condition is $0 \leq a \leq b \leq 1$.

The Borel sigma-algebra is the standard choice in continuous probability because there exists a unique probability measure on $\mathcal{B}([0,1])$ that assigns to each half-open interval $(a,b]$ its exact length $b-a$. This measure, known as Lebesgue measure, corresponds to the uniform distribution and avoids the non-measurability issues of the power set.

Conditions: The context is the continuous setting.; The measure is defined on $\mathcal{B}([0,1])$.; The endpoint condition is $0 \leq a \leq b \leq 1$.

We cannot use the power set as the sigma-algebra for a continuous uniform distribution on $[0,1]$ because there is no function defined on all subsets of $[0,1]$ that can simultaneously satisfy the length property, countable additivity, and translation invariance (assuming the axiom of choice).

Conditions: The sample space is $\Omega = [0,1]$.; We assume the axiom of choice.; The measure must satisfy length, countable additivity, and translation invariance.

We cannot use the power set as the sigma-algebra for a continuous uniform distribution on $[0,1]$ because there is no function defined on all subsets of $[0,1]$ that can simultaneously satisfy the length property, countable additivity, and translation invariance (assuming the axiom of choice).

Conditions: The sample space is $\Omega = [0,1]$.; We assume the axiom of choice.; The measure must satisfy length, countable additivity, and translation invariance.