Lebesgue Spaces

# Lebesgue Spaces

Recall from the The Set of p-Integrable and Essentially Bounded Functions page that if $(X, \mathfrak T, \mu)$ is a measure space, then for $1 \leq p < \infty$ we can define the space $\mathcal L^p (X, \mathfrak T, \mu)$ as the set of all measurable functions $f : X \to \mathbb{C}$ such that:

(1)
\begin{align} \quad \int_X |f|^p \: d \mu < \infty \end{align}

Such functions are called $p$-integrable functions, and we defined the seminorm $\| \cdot \|_p$ by:

(2)
\begin{align} \quad \| f \|_p = \left ( \int_X |f|^p \: d \mu \right )^{1/p} \end{align}

Furthermore, we can define the space $L^{\infty} (X, \mathfrak T, \mu)$ as the set of all measurable functions $f : X \to \mathbb{C}$ such that there exists an $M \in \mathbb{R}$, $M > 0$ such that $|f(x)| \leq M$ $\mu$-almost everywhere on $X$.

Such functions are called essentially bounded functions, and we defined the seminorm $\| \cdot \|_{\infty}$ by:

(3)
\begin{align} \quad \| f \|_{\infty} = \inf \{ M > 0 : |f(x)| \leq M, \: \mu-\mathrm{a.e. \: on \:} X \} \end{align}

We noted that for $1 \leq p \leq \infty$ that $\mathcal L^p(X, \mathfrak T, \mu)$ is a vector space. We would now like to show that $\mathcal L^p (X, \mathfrak T, \mu)$ is a normed vector space.

Note that indeed that for $1 \leq p \leq \infty$, $\| \cdot \|_p$ is indeed a seminorm since for all $\lambda \in \mathbb{C}$ and for all $f \in \mathcal L^p(X, \mathfrak T, \mu)$ we have that:

(4)
\begin{align} \quad \| \lambda f \|_p = | \lambda | \| f \|_p \end{align}

And from Minkowski's Inequality we have that:

(5)
\begin{align} \quad \| f + g \|_p \leq \| f \|_p + \| g \|_p \end{align}

Now let $1 \leq p \leq \infty$ and define:

(6)
\begin{align} \quad N_p = \{ f \in \mathcal L^p (X, \mathfrak T, \mu) : \| f \|_p = 0 \} \end{align}

Note that $N_p$ is a vector subspace of $\mathcal L^p (X, \mathfrak T, \mu)$ since $0 \in N_p$, and if $f, g \in N_p$ then by Minkowski's inequality $\| f + g \|_p \leq \| f \|_p + \| g \|_p = 0 + 0 = 0$ so $(f + g) \in N_p$ and if $f \in N_p$ and $\lambda \in \mathbb{C}$ then $\| \lambda f \|_p = | \lambda | \| f \|_p = | \lambda | \cdot 0 = 0$ and so $\lambda f \in N_p$. Hence the quotient space $\mathcal L^p (X, \mathfrak T, \mu) / N_p$ is a vector space where for all $f, g \in \mathcal L^p (X, \mathfrak T, \mu)$ and $\lambda \in \mathbb{C}$ we define:

(7)
\begin{align} \quad [f] + [g] = [f + g] \end{align}
(8)
\begin{align} \quad \lambda [f] = [\lambda f] \end{align}

And we define a norm for all $f \in \mathcal L^p (X, \mathfrak T, \mu)$ by:

(9)
\begin{align} \quad \| [f] \|_p = \| f \|_p \end{align}

Note that this norm is well-defined since if $[f] = [g]$ then $[f - g] = [0]$, and so $\| [f - g] \|_p = 0$. Therefore $| \| f \|_p - \| g \|_p | \leq \| f - g \|_p$ so $\| f \|_p = \| g \|_p$.

 Definition: Let $(X, \mathfrak T, \mu)$ be a vector space. For $1 \leq p \leq \infty$, the Lebesgue Space is the normed vector space defined as $L^p (X, \mathfrak T, \mu) = \mathcal L^p (X, \mathfrak T, \mu) / N_p$ with the norm $\| [f] \|_p = \| f \|_p$.

It is conventional that denote the equivalence class "$[f]$" simply by "$f$".