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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.

A sigma-algebra is closed under countable intersections because any countable intersection can be rewritten in terms of complements and countable unions using De Morgan's law. Since a sigma-algebra is already closed under complements and countable unions, it must also be closed under countable intersections.

Conditions: Each $A_k \in \mathcal{F}$.; $\mathcal{F}$ is closed under complements and countable unions.

A sigma-algebra is closed under countable intersections because any countable intersection can be rewritten in terms of complements and countable unions using De Morgan's law. Since a sigma-algebra is already closed under complements and countable unions, it must also be closed under countable intersections.

Conditions: Each $A_k \in \mathcal{F}$.; $\mathcal{F}$ is closed under complements and countable unions.

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.