Equivalence (measure theory)

In mathematics, and specifically in measure theory, equivalence is a notion of two measures being qualitatively similar. Specifically, the two measures agree on which events have measure zero. In mathematics, and specifically in measure theory, equivalence is a notion of two measures being qualitatively similar. Specifically, the two measures agree on which events have measure zero. Let μ {displaystyle mu } and ν {displaystyle u } be two measures on the measurable space ( X , A ) {displaystyle (X,{mathcal {A}})} , and let be the set of all μ {displaystyle mu } -null sets; N ν {displaystyle {mathcal {N}}_{ u }} is similarly defined. Then the measure ν {displaystyle u } is said to be absolutely continuous in reference to μ {displaystyle mu } iff N ν ⊃ N μ {displaystyle {mathcal {N}}_{ u }supset {mathcal {N}}_{mu }} . This is denoted as ν ≪ μ {displaystyle u ll mu } . The two measures are called equivalent iff μ ≪ ν {displaystyle mu ll u } and ν ≪ μ {displaystyle u ll mu } , which is denoted as μ ∼ ν {displaystyle mu sim u } . An equivalent definition is that two measures are equivalent if they satisfy N μ = N ν {displaystyle {mathcal {N}}_{mu }={mathcal {N}}_{ u }} .