Measure, Integration & Real Analysis, 2021a

202 Chapter 7 L p Spaces
7B
L p (μ)
Definition of L p (μ)
Suppose (X, S, μ) is a measure space and 1 ≤ p ≤ ∞. If there exists a nonempty set
E ∈Ssuch that μ(E) =0, then ‖χ E
‖ p = 0 even though χ E ̸= 0; thus ‖·‖ p is not a
norm on L p (μ). The standard way to deal with this problem is to identify functions
that differ only on a set of μ-measure 0. To help make this process more rigorous, we
introduce the following definitions.
7.15 Definition Z(μ); ˜f
Suppose (X, S, μ) is a measure space and 0 < p ≤ ∞.
• Z(μ) denotes the set of S-measurable functions from X to F that equal 0
almost everywhere.
• For f ∈L p (μ), let ˜f be the subset of L p (μ) defined by
˜f = { f + z : z ∈Z(μ)}.
The set Z(μ) is clearly closed under scalar multiplication. Also, Z(μ) is closed
under addition because the union of two sets with μ-measure 0 is a set with μ-
measure 0. Thus Z(μ) is a subspace of L p (μ), as we had noted in the third bullet
point of Example 6.32.
Note that if f , F ∈L p (μ), then ˜f = ˜F if and only if f (x) =F(x) for almost
every x ∈ X.
7.16 Definition L p (μ)
Suppose μ is a measure and 0 < p ≤ ∞.
• Let L p (μ) denote the collection of subsets of L p (μ) defined by
L p (μ) ={ ˜f : f ∈L p (μ)}.
• For ˜f , ˜g ∈ L p (μ) and α ∈ F, define ˜f + ˜g and α ˜f by
˜f + ˜g =(f + g)˜ and α ˜f =(α f )˜.
The last bullet point in the definition above requires a bit of care to verify that it
makes sense. The potential problem is that if Z(μ) ̸= {0}, then ˜f is not uniquely
represented by f . Thus suppose f , F, g, G ∈L p (μ) and ˜f = ˜F and ˜g = ˜G. For
the definition of addition in L p (μ) to make sense, we must verify that ( f + g)˜ =
(F + G)˜. This verification is left to the reader, as is the similar verification that the
scalar multiplication defined in the last bullet point above makes sense.
You might want to think of elements of L p (μ) as equivalence classes of functions
in L p (μ), where two functions are equivalent if they agree almost everywhere.
Measure, Integration & Real Analysis, by Sheldon Axler