Measure theory MOC

Essential supremum and infimum

The essential supremum and infinimum of a measurable function 𝑓 :𝑋 →𝑌¹ are the supremum and infimum of a function almost everywhere, measure i.e.

esssup𝑓=inf{𝐶∈𝑌:𝜇({𝑠∈𝑋:𝑓(𝑠)>𝐶})=0}essinf𝑓=sup{𝐶∈𝑌:𝜇({𝑠∈𝑋:𝑓(𝑠)<𝐶})=0}

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Footnotes

1. Typically 𝑌 =ℝ, but it may be any ordered measure space. ↩