 Probabilities are generally defined as measures in a measure space. A measure space is a set \(\Omega\) (the sample space) together with a \(\sigma\)-algebra (also called \(\sigma\)-field) \(\mathcal{A}\) over \(\Omega\), which is a non-empty set of subsets of \(\Omega\) such that \(A\in \mathcal{A}\) implies that \(\Omega-A\in \mathcal{A}\), and \(A_i\in \mathcal{A}\) for all natural numbers \(i\), implies that \(\bigcup_i A_i\in \mathcal{A}\). A measure is a function \(\mu\) defined on the \(\sigma\)-algebra \(\mathcal{A}\), such that \(\mu(A) \ge 0\) for every set \(A \in\mathcal{A}\) and \(\mu(\bigcup_i A_i) = \sum_i\mu(A_i)\) whenever \(A_i\cap A_j = \emptyset\) for each \(i,j\).