proof of existence of the essential supremum


Suppose that (Ω,ℱ,μ) is a σ-finite measure space and 𝒮 is a collectionMathworldPlanetmath of measurable functionsMathworldPlanetmath f:Ω→ℝ¯. We show that the essential supremumMathworldPlanetmath of 𝒮 exists and furthermore, if it is nonempty then there is a sequence fn∈𝒮 such that

ess⁢sup⁡𝒮=supn⁡fn.

As any σ-finite measureMathworldPlanetmath is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to a probability measure (http://planetmath.org/AnySigmaFiniteMeasureIsEquivalentToAProbabilityMeasure), we may suppose without loss of generality that μ is a probability measure. Also, without loss of generality, suppose that 𝒮 is nonempty, and let 𝒮′ consist of the collection of maximums of finite sequences of functions in 𝒮. Then choose any continuousMathworldPlanetmathPlanetmath and strictly increasing θ:ℝ¯→ℝ. For example, we can take

θ⁢(x)={x/(1+|x|),if ⁢|x|<∞,1,if ⁢x=∞,-1,if ⁢x=-∞.

As θ⁢(f) is a boundedPlanetmathPlanetmathPlanetmath and measurable function for all f∈𝒮′, we can set

α=sup⁡{∫θ⁢(f)⁢𝑑μ:f∈𝒮′}.

Then choose a sequence gn in 𝒮′ such that ∫θ⁢(gn)⁢𝑑μ→α. By replacing gn by the maximum of g1,…,gn if necessary, we may assume that gn+1≥gn for each n. Set

f=supn⁡gn.

Also, every gn is the maximum of a finite sequence of functions gn,1,…,gn,mn in 𝒮. Therefore, there exists a sequence fn∈𝒮 such that

{f1,f2,…}={gn,m:n≥1,1≤m≤mn}.

Then,

f=supn⁡fn.

It only remains to be shown that f is indeed the essential supremum of 𝒮. First, by continuity of θ and the dominated convergence theorem,

∫θ⁢(f)⁢𝑑μ=limn→∞⁡∫θ⁢(gn)⁢𝑑μ=α.

Similarly, for any g∈𝒮,

∫θ⁢(f∨g)=limn→∞⁡∫θ⁢(gn∨g)⁢𝑑μ≤α.

It follows that θ⁢(f∨g)-θ⁢(f) is a nonnegative function with nonpositive integral, and so is equal to zero μ-almost everywhere. As θ is strictly increasing, f∨g=f and therefore f≥g μ-almost everywhere.

Finally, suppose that g:Ω→ℝ¯ satisfies g≥h (μ-a.e.) for all h∈𝒮. Then, g≥fn and,

g≥supn⁡fn=f

μ-a.e., as required.

Title proof of existence of the essential supremum
Canonical name ProofOfExistenceOfTheEssentialSupremum
Date of creation 2013-03-22 18:39:25
Last modified on 2013-03-22 18:39:25
Owner gel (22282)
Last modified by gel (22282)
Numerical id 7
Author gel (22282)
Entry type Proof
Classification msc 28A20