The difference between Measure space and Σ-algebra
When used as nouns, measure space means a measurable space which has a positive measure defined on its σ-algebra, whereas σ-algebra means a collection of subsets of a given set, such that the empty set is part of this collection, the collection is closed under complements (with respect to the given set) and the collection is closed under countable unions.
check bellow for the other definitions of Measure space and Σ-algebra
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Measure space as a noun (analysis):
A measurable space which has a positive measure defined on its σ-algebra.
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Σ-algebra as a noun (analysis):
A collection of subsets of a given set, such that the empty set is part of this collection, the collection is closed under complements (with respect to the given set) and the collection is closed under countable unions.