Let μ be a measure on a sigma algebra Σ of subsets of a setX. An element A in Σ is said to have measure zero if μ(A)=0.
Any set of measure zero is a null set. The opposite is not true, because a null set is not required to be measurable, that is, to be an element in Σ. However, any null set is a subset of a set of measure zero.
Measure informs analysis and is one of the key building blocks of the modern theory of analysis and probability.
Measure Property 2: The value of μ under any finite or countably infinite disjoint union of subsets of X that are also elements of M is equal to the finite or countably infinite sum respectively of the value of μ under each of the subsets.
Measure allows the expansion of the definition of the integral to functions whose domain is any arbitrary set with a corresponding σ-algebra and measure.
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