In mathematics, the concepts of essential infimum and essential supremum are related to the notions of infimum and supremum, but adapted to measure theory and functional analysis, where one often deals with statements that are not valid for all elements in a set, but rather almost everywhere, i.e., except on a set of measure zero.

While the exact definition is not immediately straightforward, intuitively the essential supremum of a function is the smallest value that is larger than or equal to the function values everywhere when allowing for ignoring what the function does at a set of points of measure zero.

For example, if one takes the function $ f(x) $ that is equal to zero everywhere except at $ x = 0 $ where $ f(0) = 1 $, then the supremum of the function equals one. However, its essential supremum is zero because we are allowed to ignore what the function does at the single point where $ f $ is peculiar. The essential infimum is defined in a similar way.

$ L^∞ $ is a function space. Its elements are the essentially bounded measurable functions. More precisely, $ L^∞ $ is defined based on an underlying measure space $ (S, Σ, μ) $.

Start with the set of all measurable functions from $ S $ to $ \mathbb{R} $ which are essentially bounded, i.e. bounded up to a set of measure zero. Two such functions are identified if they are equal almost everywhere. Denote the resulting set by $ L^∞(S, μ) $.

One application of $ ℓ^∞ $ and $ L^∞ $ is in economies with infinitely many commodities. In simple economic models, it is common to assume that there is only a finite number of different commodities, e.g. houses, fruits, cars, etc., so every bundle can be represented by a finite vector, and the consumption set is a vector space with a finite dimension.

But in reality, the number of different commodities may be infinite. For example, a "house" is not a single commodity type since the value of a house depends on its location. So the number of different commodities is the number of different locations, which may be considered infinite. In this case, the consumption set is naturally represented by $ L^∞. $

• Lp spaces and Lebesgue integrals
• L-infinity#Function space

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• Template:Functional analysis
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Contents:
• 1 Sequence space
• 2 Function space
• 3 Applications
• 4 References
• 5 Template:Functional analysis

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