MAT3400/4400 — Supplementary Notes
Lp vs Lp
Throughout this note we fix a measure space (Ω, A, µ). We first recall some definitions
and results from [MW13].
Definition 1. Let p ∈ (0, ∞). The set of p-integrable functions is defined to be
Z
p
p
L (Ω, A, µ) = f : Ω → C f is A-measurable and
|f | dµ < ∞ .
Ω
For an A-measurable function f we define the p-norm of f as
1
p
|f | dµ
.
Z
p
kf kp =
Ω
Note that the p-integrable functions are exactly those functions for which kf kp < ∞.
Definition 2. For an A-measurable function f we define the ∞-norm of f (also called
the essential supramum of f ) by
kf k∞ = inf {M ∈ R∗ | |f | ≤ M µ-a.e.} .
The set of essentially bounded functions is defined to be
L∞ (Ω, A, µ) = {f : Ω → C | f is A-measurable and kf k∞ < ∞} .
Proposition 3. Let p ∈ [1, ∞]. Then Lp (Ω, A, µ) is a linear space and the p-norm
defines a seminorm on Lp (Ω, A, µ).
Proof. By [MW13, Proposition 13.4b] Lp (Ω, A, µ) is a linear space and [MW13, Proposition 13.4, Theorem 13.10] show that k · kp is a seminorm on Lp (Ω, A, µ).
Lemma 4. Let p ∈ [1, ∞] and let f : Ω → C be A-measurable. Then kf kp = 0 if and
only if f = 0 µ-a.e..
Proof. Suppose that p < ∞. The case of p = ∞ is left as an exercise for the reader. By
[MW13, Exercise 5.52] we get that
Z
kf kp = 0 ⇐⇒
|f |p dµ = 0 ⇐⇒ |f |p = 0 µ-a.e. ⇐⇒ f = 0 µ-a.e..
Ω
Whenever (Ω, A, µ) is such that there are functions that are 0 almost everywhere without
being 0 everywhere, the above shows that k · kp is not a norm. To remedy this, we will
identify functions that are equal almost everywhere. Formally this is done by introduce
an equivalence relation on Lp (Ω, A, µ) and then taking a quotient space.
1
MAT3400/4400 — Supplementary Notes
Definition 5. Let p ∈ [1, ∞]. We say that two functions f, g ∈ Lp (Ω, A, µ) are equivalent
if f = g µ-a.e.. We write
f ∼ g ⇐⇒ f = g µ-a.e..
Note that by Lemma 4 f ∼ g if and only if kf − gkp = 0.
Lemma 6. Let p ∈ [1, ∞]. The relation defined in Definition 5 is an equivalence relation.
Proof. We have to check that ∼ is reflexive, symmetric, and transitive. That it is reflexive
and symmetric is obvious. To show transitivity let f, g, h ∈ Lp (Ω, A, µ) be such that
f ∼ g and g ∼ h. Then,
kf − hkp = kf − g + g − hkp ≤ kf − gkp + kg − hkp = 0
So f ∼ h.
We will write [f ] for the equivalence class of a function f ∈ Lp (Ω, A, µ). That is
[f ] = {g ∈ Lp (Ω, A, µ) | f ∼ g} .
Definition 7. Let p ∈ [1, ∞] and let ∼ be the equivalence relation from Definition 5.
Define Lp (Ω, A, µ) to be the quotient space Lp (Ω, A, µ)/ ∼. That is
Lp (Ω, A, µ) = {[f ] | f ∈ Lp (Ω, A, µ)} .
Lemma 8. Let p ∈ [1, ∞]. Under the operations [f ] + [g] = [f + g] and α[f ] = [αf ],
f, g ∈ Lp (Ω, A, µ), α ∈ C, Lp (Ω, A, µ) is a linear space.
Proof. First we must verify that the operations are indeed well defined, that is that if
f, g, f 0 , g 0 ∈ Lp (Ω, A, µ) and f ∼ f 0 , g ∼ g 0 then f + g ∼ f 0 + g 0 . We see that
k(f + g) − (f 0 − g 0 )kp = k(f − f 0 ) + (g − g 0 )kp ≤ kf − f 0 kp + kg − g 0 kp = 0 + 0.
So the addition is well defined. A similar but simpler computation shows that scalar
multiplication is well defined.
It is tedious, but not hard, to check that operations satisfy all the vector space axioms.
Lemma 9. Let p ∈ [1, ∞). The p-norm given by
Z
k[f ]kp = kf kp =
Ω
defines a norm on Lp (Ω, A, µ).
2
1
p
|f | dµ
,
p
MAT3400/4400 — Supplementary Notes
Proof. Again we first verify that the norm is well defined. Suppose f ∼ g. Since f = g
µ-a.e., we have
Z
1 Z
1
p
p
p
p
kf kp =
|f | dµ
=
|g| dµ
= kgkp .
Ω
So the p-norm on
Lp (Ω, A, µ)
Ω
is well defined.
Using that the p-norm on Lp (Ω, A, µ) is a seminorm, we get that the p-norm on Lp (Ω, A, µ)
is a seminorm. It then follows from Lemma 4 that it is in fact a norm.
When working with Lp we will often abuse notation and simply talk about functions in
Lp rather than equivalence classes of functions. As long as our discussion is limited to
integrals this should not cause too much confusion. But be aware that this can be tricky,
for instance it does not make sense to say that a function in Lp is continuous, since f ∼ g
and f continuous does not imply that g is continuous (can you give an example?).
References
[MW13] John N. McDonald and Neil A. Weiss. A course in real analysis. Academic
Press, Inc., San Diego, CA, second edition, 2013. Biographies by Carol A.
Weiss.
3