I'm on a quest to learn about operator algebras in the hopes of understanding the many interesting ways they have been applied to quantum field theory. This note is the first in a series that will build the essential aspects of the theory from the ground up. I will not prove everything, giving references for the proofs of many lemmas, but I will try to give enough detail that the mathematical underpinnings of the theory are clear.

This post is about $L^p$ spaces. These are Banach spaces of functions — with the $L^{\infty}$ space actually being a Banach algebra — that show up frequently in the theory of general operator algebras. I will introduce the basics of measure theory and the general theory of Lebesgue integration, introduce the $L^p$ spaces, and prove some basic theorems concerning them.

The outline is:

- In
**section 1**, I will introduce the basic tools of measure theory: $\sigma$-algebras, measurable functions, and measures. - In
**section 2**, I will discuss some elementary properties of real- and complex-valued measurable functions. - In
**section 3**, I will define the Lebesgue integral on a general measure space and state some important theorems such as the dominated convergence theorem. - In
**section 4**, I will make some comments on the role of sets of measure zero, and how to think about them. - In
**section 5**, I will introduce the $L^p$ spaces and prove that they are Banach spaces. I will also show that $L^2$ is a Hilbert space.

**Prerequisites: **Basics of topology and real analysis.

__Table of Contents__

- $\sigma$-algebras, measurability and measures
- Real and complex measurable functions
- Integration
- Sets of measure zero
- $L^p$ spaces

## 1. $\sigma$-algebras, measurability, and measures

*some*subsets of $X$ should be considered measurable. But when we think about how crazy sets can be — completely arbitrary collections of points in $X$ — it shouldn't be so surprising that a consistent theory of measure might only allow us to measure a restricted family of subsets.

**measure**, and which we will think of as assigning volumes to elements of $E$.

**$\sigma$-algebra**on $X$. Once a $\sigma$-algebra has been specified, $X$ is said to be a

**measurable space**and the sets in the $\sigma$-algebra are said to be

**measurable sets**. Standard manipulations in set theory show that conditions (i-iii) imply that $\Sigma$ contains $X = \varnothing^c,$ is closed under countable intersections $\cap_j E_j = (\cup_j E_j^c)^c$, and is closed under set subtractions $A, B \in \Sigma \Rightarrow A - B = A \cap B^c.$

**Borel algebra**on $X$ is the $\sigma$-algebra generated by open subsets of $X$.

**measurable function**if preimages of open sets in $Y$ are measurable in $X$; this mimics the definition of a continuous function on a topological space, for which preimages of open sets are open. Note that if $X$ is a topological space, then all continuous maps from $X$ to $Y$ are measurable with respect to the Borel algebra.

**measure**is a map $\mu : \Sigma \to [0, \infty]$ satisfying $\mu(\varnothing) = 0$ and $\mu(\cup_j E_j) = \sum_j \mu(E_j)$ for pairwise-disjoint measurable sets $E_j.$ From this definition, many interesting properties can be proved; the most important of these, for the moment, is that measures are monotonic: for $A \subseteq B$ both measurable, we have $\mu(B) = \mu(A) + \mu(B - A) \geq \mu(A).$

**measure space**.

## 2. Real and complex measurable functions

- If $u : X \to \mathbb{R}$ and $v : X \to \mathbb{R}$ are measurable, then $f(x) = u(x) + i v(x)$ is measurable.
- If $f : X \to \mathbb{C}$ is measurable, then its real part, its complex part, and its magnitude are all measurable.
- If $f $ and $g$ are measurable functions from $X$ to $\mathbb{R}$ or $\mathbb{C}$, then $fg$ and $f+g$ are measurable.
- The pointwise supremum and infimum (and hence limit-superior and limit-inferior) of any sequence of measurable functions are measurable.

**positive and negative parts**$f^+ = \max\{f, 0\}$ and $f^{-} = - \min\{f, 0\}$ are measurable. The decomposition of $f$ into positive and negative parts will play an important part in defining integration in the next section.

**characteristic function**$\chi_E$ to be the function from $X$ to $\mathbb{R}$ that takes the value $1$ on $E$ and $0$ on $E^c.$ Clearly $E$ is a measurable set if and only if $\chi_E$ is a measurable function.

## 3. Integration

**simple function**is a measurable function whose range has only finitely many values. Any such function can be written as a finite sum of characteristic functions over disjoint measurable sets $E_j$:

- $0 \leq f \leq g$ implies $\int_{X} d\mu\, f \leq \int_{X} d\mu\, g.$
- $A \subseteq B$ and $f \geq 0$ implies $\int_{A} d\mu\, f \leq \int_{B} d\mu\, f.$
- $c \in [0, \infty)$ and $f \geq 0$ implies $\int_X d\mu\, c f = c \int_X d\mu\, f.$
- $\mu(A) = 0$ or $f|_A = 0$ both imply $\int_{X} d\mu\, f = 0.$

- Integration is linear: $f, g \geq 0$ implies $\int_{X} d\mu\, (f + g) = \int_{X} d\mu\, f + \int_{X} d\mu\, g.$
- If $f \geq 0$ is a measurable function on $X$, then the map $E \mapsto \int_{E} d\mu\, f$ is a measure on $X$.

**integrable**if $\int_{X} d\mu\, |f|$ is finite. The space of all such functions is denoted $\mathcal{L}^1(\mu).$ The real and complex parts of an integrable function are also integrable, as are the positive and negative parts of those. This lets us define the integral of a general function in $\mathcal{L}^1(\mu)$:

**dominated convergence theorem**. I won't prove it; the proof isn't hard, but it requires a few lemmas that I don't think are particularly instructive on their own. The statement is that if $f_n$ is a sequence of complex, measurable functions on $X$ that converge pointwise to a function $f,$ and that satisfy $|f_n| \leq g$ for some $g \in \mathcal{L}^1(\mu)$, then $f_n$ and $f$ are also in $\mathcal{L}^1(\mu)$ and the integral of the sequence converges:

## 4. Sets of measure zero

## 5. $L^p$ spaces

**p-norm**of $f$ by

**p-norm**" is slightly inaccurate, as $\lVert \cdot \rVert_p$ isn't actually a norm; it isn't positive definite, since it assigns zero to all functions that vanish almost everywhere. We define $\mathcal{L}^p(\mu)$ to be the set of functions on which the p-norm is finite, and define $L^p(\mu)$ to be the quotient of $\mathcal{L}^p(\mu)$ by the space of functions that vanish almost everywhere. We will see that the p-norm is an actual norm on $L^p(\mu).$ To show this, it suffices to show that the p-norm is a

*seminorm*on $\mathcal{L}^p(\mu),$ i.e., that it satisfies all the properties of a norm except for positive definiteness.

**essential supremum**of a measurable function $f : X \to [0, \infty]$ to be the smallest number $a$ such that the set $\{x | f(x) > a\}$ has measure zero. Formally, we write

**exists**in the first place; but this follows from Holder's inequality, which gives $\lVert \bar{f} g \rVert_1 \leq \lVert \bar{f} \rVert_2 \lVert g \rVert_2 < \infty.$ This inner product clearly induces the $2$-norm, and is linear in $g,$ and is positive definite. The only thing we need to check is that it satisfies $\langle f | g \rangle = \overline{\langle g | f \rangle},$ but this follows readily from considering real and imaginary parts of integrals.

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