---
title: The L<sup>p</sup> Spaces
parent: Lebesgue Integral
grand_parent: Measure and Integration
nav_order: 4
---

# {{ page.title }}

{% definition %}
Let $(X,\mathcal{A},\mu)$ be a measure space and let $p \in [1,\infty)$.
We write $\mathscr{L}^p(X,\mathcal{A},\mu)$ for the set of all
measurable functions $f : X \to \KK$ such that $\abs{f}^p$ is integrable.
For such $f$ we write

$$
\norm{f}_p = {\bigg\lparen\int_X \abs{f}^p \, d\mu\bigg\rparen}^{\!1/p}.
$$
{% enddefinition %}

{% proposition %}
Endowed with pointwise addition and scalar multiplication
$\mathscr{L}^p(X,\mathcal{A},\mu)$ becomes a vector space.
{% endproposition %}

{% proof %}
We show that $\mathscr{L}^p := \mathscr{L}^p(X,\mathcal{A},\mu)$ is a linear subspace of
the vector space of all $\KK$-valued functions on $X$.
The set $\mathscr{L}^p$ is nonempty since
it contains the zero function.
Now, suppose $f$ and $g$ are in $\mathscr{L}^p$.
Then the sum $f+g$ is measurable, because $f$ and $g$ are measurable.
Moreover, the function $\abs{f+g}^p$ is integrable, because we have the estimate

$$
\abs{f+g}^p
\le (\abs{f} + \abs{g})^p
\le \big\lparen 2 \max(\abs{f},\abs{g}) \big\rparen^p
\le 2^p (\abs{f}^p + \abs{g}^p),
$$

where $\abs{f}^p$ and $\abs{g}^p$ are integrable.
This proves that $f+g$ lies in $\mathscr{L}^p$.
Finally, it is easy to see that $\alpha f$ lies in $\mathscr{L}^p$
for any scalar $\alpha \in \KK$.
{% endproof %}

{% proposition %}
$\norm{\cdot}_p$ is a seminorm on $\mathscr{L}^p(X,\mathcal{A},\mu)$.
{% endproposition %}

{% theorem * Young Inequality %}
Consider $p,q > 1$ such that $1/p + 1/q = 1$. Then

$$
a \cdot b \le \frac{a^p}{p} + \frac{b^q}{q} \qquad \forall a,b \ge 0.
$$
{% endtheorem %}