The Lebesgue Space of Power-Integrable Functions

In the setting of Measure Spaces and the Lebesgue Integral: Standing Notation, let $(X,\mathcal{F},\mu)$ be a measure space, let $p$ be a real number with $1\le p$, and write $\mathcal{L}^{p}$ for the set of $p$-integrable functions on $(X,\mathcal{F},\mu)$ and $\lVert\cdot\rVert_{p}$ for the $p$-seminorm.

1. (Almost-everywhere equality)¶ For $f,g\in\mathcal{L}^{p}$ write $f\sim g$ when $f(x)=g(x)$ for almost every $x\in X$. This is an equivalence relation on $\mathcal{L}^{p}$: it is reflexive because the empty set is null, it is symmetric because the set where $f$ and $g$ differ does not depend on their order, and it is transitive because the set where $f$ and $h$ differ is contained in the union of the set where $f$ and $g$ differ and the set where $g$ and $h$ differ, which is null by The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §null-union. The equivalence class of $f$ is written $[f]$.

2. (The Lebesgue space)¶ $L^{p}(X,\mathcal{F},\mu)$ is the set of equivalence classes of the relation of clause 1, equipped with the operations

the sum and scalar multiple on the right being the pointwise ones. These operations are well defined: $f+g$ and $cf$ lie in $\mathcal{L}^{p}$ by Minkowski's Inequality and the Seminormed Space of Power-Integrable Functions §vector-space, and if $f\sim f'$ and $g\sim g'$ then the set where $f+g$ and $f'+g'$ differ is contained in the union of the set where $f$ and $f'$ differ and the set where $g$ and $g'$ differ, hence is null by The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §null-union, while the set where $cf$ and $cf'$ differ is contained in the set where $f$ and $f'$ differ, hence is null by The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §null-union as well.

3. (The norm)¶ For $f\in\mathcal{L}^{p}$ put $\lVert[f]\rVert_{p}=\lVert f\rVert_{p}$. This does not depend on the representative, by Elementary Properties of the p-Seminorm §almost-everywhere, so it assigns a nonnegative real number to each element of $L^{p}(X,\mathcal{F},\mu)$.

4. (Notational convention)¶ An element of $L^{p}(X,\mathcal{F},\mu)$ is denoted by the same symbol as a representative of it: one writes $f$ for $[f]$ and $\lVert f\rVert_{p}$ for $\lVert[f]\rVert_{p}$ whenever the statement being made does not depend on the choice of representative.