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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.