Measure theory MOC

Essential supremum and infimum

The essential supremum and infinimum of a measurable function $𝑓 : 𝑋 \to 𝑌$ ¹ are the supremum and infimum of a function almost everywhere, #m/def/measure i.e.

e s s s u p 𝑓 = i n f {𝐶 \in 𝑌 : 𝜇 ({𝑠 \in 𝑋 : 𝑓 (𝑠) > 𝐶}) = 0} e s s i n f 𝑓 = s u p {𝐶 \in 𝑌 : 𝜇 ({𝑠 \in 𝑋 : 𝑓 (𝑠) < 𝐶}) = 0}

#state/tidy | #lang/en | #SemBr

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