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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.