\part{Measure functions}


\chapter{$\sigma$-algebrae}


\section{$\sigma$-algebra}
If you've studied general topology, you may recall that open sets form a particular family of sets called the topology. It was inspired by the properties of open sets generated by metric spaces.

In the context of measure theory, we'll eventually do a similar thing; we'll see that \emph{measurable sets} form a family of sets known as the $\sigma$-algebra. We define what a $\sigma$-algebra is so that we have familiarity once we encounter it down the track.


\begin{definition}
A \emph{$\sigma$-algebra} is a family of sets $\Sigma$
\[ \emptyset, X \in \Sigma\]
\[ \sigma \in \Sigma \implies X \setminus \sigma \in \Sigma\]
\[ \bigcup^{\infty}_{n=1} \sigma_{n} \in \Sigma \]
\end{definition}

De Morgan's law allows us to view that $\sigma$-algebrae are also closed under countable intersections.
\begin{proposition}
A \emph{$\sigma$-algebra} is a family of sets $\Sigma$
\[ \bigcap^{\infty}_{n=1} \sigma_{n} \in \Sigma \]
\end{proposition}


\section{Borel $\sigma$-algebra}

It is best to come back to this section after having read the rest of this part, since the motivation for Borel $\sigma$-algebrae comes from a desire to combine measure theory with topology, but thus far we have covered no measure theory.

I've hinted to the fact that $\sigma$-algebrae provide the magic needed to measure sets; imagine that we have a topological space and we want to measure all our open sets. We take the open sets and take countable intersections and unions to generate it into a $\sigma$-algebra of sets we can apply measure theory to; this is called the \emph{Borel $\sigma$-algebra}.


\begin{definition}[Borel $\sigma$-algebra]
Given a topological space $(X,\mathcal{T})$, the \emph{Borel $\sigma$-algebra of $(X,\mathcal{T})$} is the $\sigma$-algebra $\mathcal{B}_{\mathcal{T}}$ generated by the sets open in $(X,\mathcal{T})$.
Sets of the Borel $\sigma$-algebra are called \emph{Borel sets of $(X,\mathcal{T})$}.
\end{definition}



\begin{proposition}
Sets closed in $(X,\mathcal{T})$ are Borel sets of $(X,\mathcal{T})$.
\end{proposition}


Since the Borel $\sigma$-algebra is a $\sigma$-algebra generated by open sets (which are already Lebesgue measurable for the Euclidean topology on $\mathbb{R}$), the following proposition naturally follows.
\begin{proposition}
Let $(\mathbb{R},\mathcal{T})$ be the Euclidean topology on $\mathbb{R}$. The Borel sets of $(\mathbb{R},\mathcal{T})$ are measurable.
\end{proposition}









\chapter{Lebesgue measure}


THe concept of cardinality gives us a way of measuring the 'size' of sets, but it doesn't always capture the .

Although the notion of cardinality is extremely useful in some contexts, it fails to describe the notion of 'size' in others. Take the intervals $[0,1]$ and $[3,9]$. Their cardinalities are both 'infinite', but in another sense, $[0,1]$ has a 'length' of 1 and $[3,9]$ has a 'length' of 6. This notion of size is called the \emph{Lebesgue measure}, and is useful in mathematical analysis, specifically integration. 

The Lebesgue measure as we will define it will be able to assign a 'measure' to more complicated sets on $\mathbb{R}$ rather than just intervals (we can generalize the Lebesgue measure to $\mathbb{R}^n$, but this part will only consider the Lebesgue measure on sets of $\mathbb{R}$). However we will eventually ask if all sets of $\mathbb{R}$ can be assigned a measure according to the Lebesgue measure. Spoiler alert; no.


The ultimate goal we have is to define a general framework for ways of 'measuring' the size of sets in different ways, including our Lebesgue measure. However like much of mathematics, we need to understand concrete examples before we abstractify them.

Those who have read my book on general topology recall that I had to begin with metric spaces before I could generalize them to topological spaces. This book will similarly start with the study of the Lebesgue measure and later generalize to generic measure functions. Though we're ambitious, we still need to crawl before we learn to climb.


\section{Vitali sets}


We have an idea of some properties that our Lebesgue measure should obey, so let's idealize some essential properties for the Lebesgue measure $\lambda$.
\[\]
\[\]
\[\]

We now introduce a rather pathological type of set.
\begin{definition}[Vitali set]
\end{definition}

When we officially define our Lebesgue measure, it will obey these properties; We'll take this on faith for now. Assuming that our Lebesgue measure obeys these properties, we're disillusioned with the following discovery.

\begin{proposition}[Vitali's theorem]
Let $V$ be a Vitali set and $\lambda$ be the Lebesgue measure, then $\lambda( V )$ is undefined.
\end{proposition}

The Vitali sets are therefore an example of sets that we won't be able to measure with the Lebesgue measure. If we surrender the i, j or kth of these idealized properties (or more radically, the axiom of choice) we can have every set of $\mathbb{R}$ to be measurable, however dropping any of those properties would make the Lebesgue measure much less of a powerful tool. Under this premise, Lebesgue decided to accept the reality of non-measurable sets and continue his theory.

Aware of the limitation of our Lebesgue measure, we're prompted to give a constructive definition of the Lebesgue measure and classify what sets the Lebesgue measure is well defined on.

\section{Constructing the Lebesgue measure}
 


We'll start our journey to costruct such a function; we've got the intuition of how we want to measure intervals, so we can define the \emph{interval length function} as a precursor to the Lebesgue measure.

\begin{definition}[Interval length function]
The \emph{interval length function} is the function $\ell$ defined 
\[ \ell ([a,b]) = \ell ((a,b))=b-a\]
\end{definition}

To handle sets that aren't necessarily intervals, we consider collections of open intervals that cover the set in question and take the cover with the smallest measure to be the \emph{Lebesgue outer measure} of that set. We're essentially 'shrinking in' on the set from the outside by means of open intervals.


\begin{definition}[Lebesgue outer measure]
The \emph{Lebesgue outer measure} is the function $\lambda^{*} : \mathcal{P}(\mathbb{R}) \to [0,\infty) \cup \{+\infty\} $ defined as such.
\[ \lambda^{*}(E) = \inf \{ \sum_{n=1}^{\infty} \ell (I_n) : (I_n)_{n \in \mathbb{N}} \text{ is a sequence of open intervals } \land  E \subseteq \bigcup_{n=1}^{\infty} I_n  \} \]
Note that the sequence of $I_n$ may be finite or countable (as like any sequence).
\end{definition}


It's worth mentioning that we have permitted the domain of the Lebesgue outer measure to be the nonnegative extended real numbers; we will allow sets to be given the measure $+\infty$. This is because the sum in this definition may possibly diverge, and it will allow arithmetic of the extended real numbers to be conducted on the Lebesgue measure.

This is almost the Lebesgue measure, however $\lambda^{*}$ is well defined for any real set and in terms of countable additivity, the following proposition is the best that can be done.

\begin{proposition}
Let $(E_n)_{n \in \mathbb{N}}$ be a sequence of subsets of $\mathbb{R}$, then the following holds.
\[ \lambda^{*}( \bigcup^{\infty}_{n=1}E_n) \leq \sum^{\infty}_{n=1} \lambda^{*}( E_n) \]
\end{proposition}

With the Lebesgue outer measure, this proposition cannot be promoted to an equality, since that would make $\lambda^{*}$ exactly obey those 'idealized' Lebesgue measure properties, but then there would be sets undefined for the Lebesgue outer measure, which is a contradiction.

We still want countable additivity though, so this means that we want to restrict the domain of $\lambda^{*}$ from $\mathcal{P}(\mathbb{R})$ to $\{ \bigcap^{\infty}_{n=1} E_n \in \mathcal{P}(\mathbb{R}) : (E_n)_{n \in \mathbb{N}} \text{ are pairwise disjoint } \implies  \lambda^{*}( \bigcup^{\infty}_{n=1}E_n) = \sum^{\infty}_{n=1} \lambda^{*}( E_n) \}$.

An alternative but equivalent way to characterize this is through \emph{Carathéodory criterion}.

\begin{theorem}[Carathéodory criterion]
\end{theorem}

So the Lebesgue outer measure restricted to sets obeying the Carathéodory criterion is exactly what we want out Lebesgue measure to be; we can now define the Lebesgue measure.


\begin{definition}[Lebesgue measure]
The \emph{Lebesgue measure} is the function $\lambda : \Sigma \subset \mathcal{P}(\mathbb{R}) \to [0,\infty] $ defined as below, where elements $E \in \Sigma$ obey Carathéodory criterion.
\[ \lambda(E) = \lambda^{*}(E)\]
\end{definition}

%We're using $\Sigma$ to represent all the sets that the Lebesgue measure is well defined on. We'll find a more formal characterization of what this is later.


\begin{definition}[Lebesgue measurable set]
A subset $E$ of $\mathbb{R}$ is a \emph{Lebesgue measurable set} iff $E \in \mathrm{dom}(\lambda)$
\end{definition}


\begin{proposition}
Open sets in the Euclidean topology on $\mathbb{R}$ are Lebesgue measurable. In other words, $\mathcal{B}_{\mathcal{T}} \subseteq \mathrm{dom}(\lambda)$.
\end{proposition}



It's interesting to note that we could also define the Lebesgue measure by considering a \emph{Lebesgue inner measure}, and taking the sets where the inner and outer measures are equal; this was Lebesgue's original idea.

\begin{proposition}
$E \subseteq \mathbb{R}$ is Lebesgue measurable iff the following holds.
\[ \lambda^{*}(E)= \lambda_{*}(E) \]
\end{proposition}

\begin{proposition}
Let $I \subseteq \mathbb{R}$ be an interval, then $I$ is Lebesgue measurable.
\end{proposition}


\begin{proposition}
For any $y\in \mathbb{R}$ we have the following.
\[ \lambda(E) = \lambda( \{ x+e : e\in E\} ) \]
\end{proposition}


\begin{proposition}
$\mathrm{dom}(\lambda)$ is a $\sigma$-algebra.
\end{proposition}


Lebesgue null set
countable sets are lebesgue null sets
Cantor set is a lebesgue null set


\chapter{Measures}

Now that we have an concrete understanding of one type of measure function, we propose a definition for a generic measure function; this will allow us to apply measure theory to diverse and abstract settings.

\begin{definition}[Measurable space]
A \emph{measurable space} is an ordered pair $(X,\Sigma)$. It represents a set with potential for a well defined measure defined on $\Sigma$.
\begin{itemize}
\item $X$ is a space (set)
\item $\Sigma$ is a $\sigma$-algebra on $X$
\end{itemize}
\end{definition}

\begin{definition}[Borel space]
Let $(X,\mathcal{T})$ be a topological space, then the \emph{Borel space of $(X,\mathcal{T})$} is the measurable space $(X,\mathcal{B}_{\mathcal{T}}$.
\end{definition}


Measurable spaces have potential for a measure to be endowned onto it, while a measure space is a measurable space with a specific measure attached.

\begin{definition}[Measure space]
A \emph{measure space} is a 3-tuple $(X,\Sigma,\mu)$. It represents a measurable space equipped with a specific measure on its $\Sigma$.
\begin{itemize}
\item $X$ is a space (set)
\item $\Sigma$ is a $\sigma$-algebra on $X$
\item $\mu$ is a measure on $\Sigma$
\end{itemize}
\end{definition}

We define measurable spaces as their own object since there are things that we can say about a sets measurability without the need to involve specific measures; so long as our chosen $\sigma$-algebra includes the set, any measure that could be applied to that measurable space will give that set a measure.

\begin{definition}[Measure]
A \emph{measure on $(X,\Sigma)$} is a function $\mu : \Sigma \to [0,\infty)\cup\{+\infty\}$ that satisfies the following.
\begin{itemize}
	\item $ \mu(\emptyset) = 0 $
	\item $\mu ( \bigcup_{i \in \mathbb{R}} E_i ) =  \sum{i \in \mathbb{R}}  \mu ( E_i )$
\end{itemize}
\end{definition}


\begin{definition}[measurable set]
$E$ is a \emph{measurable set of $(X,\Sigma)$} iff $E \in \Sigma$
\end{definition}
Note that this doesn't depend on any measure, but rather the $\sigma$-algebra which determines the 'measurability structure' chosen for $X$

The idea of a null set however is dependent on the chosen measure.

\begin{definition}[null set]
$E$ is a \emph{null set of $(X,\Sigma,\mu)$} iff $\mu (E) = 0$. When the measurable space is known but there may be multiple measures, this may be called a $\mu$-null set.
\[E \text{ is a null set of } (X,\Sigma,\mu) \iff \mu(E)=0\]
\end{definition}

It's obvious that all null sets are measurable (since by definition they have a measure, specifically 0).

\begin{definition}[Almost everywhere property]
A property $P$ holds \emph{almost everywhere on $(X,\Sigma,\mu)$} iff the set of elements on which it doens't hold is a null set.
\end{definition}

- complete measure space

\begin{definition}[Absolute continuity]
Let $(X,\sigma)$ be a measurable space compatible with measures $\mu,\nu$. Then \emph{$\mu$ is absolutely continuous with respect to $\nu$} iff every $\nu$-null set is a $\mu$-null set.
\end{definition}





\chapter{Classes of measure functions}

Now that we have defined measures in a general sense, it is perhaps enlightening to imagine different types of measures with different properties.

\section{Finite measures}
Recall that we defined the codomain of measures to be the nonnegative extended real numbers, however some measures may not even require sets of infinite measure. These are called \emph{finite measures} and they often behave nicer than generic measures.
- finite measure
\begin{definition}[Finite measure]
Let $(X,\Sigma)$ be a measurable space that $\mu$ is well defined on, a \emph{finite measure} is a measure function $\mu$ that maps no set of $\Sigma$ to infinity.
\[ \mu \text{ is finite } \iff \forall E \in \Sigma [ \mu (E) < \infty ] \]
\end{definition}

\section{Atomic measures}
\begin{definition}[Atom]
Let $(X,\Sigma)$ be a measurable space that $\mu$ is well defined on, an \emph{atom} is a measurable set $A$ with positive measure and all its measurable subsets are either null sets or have the same measure as $A$.
If $\mu$ contains an atom, it is called an \emph{atomic measure}.
\[ A \text{ is an atom of } \mu \iff \mu(A) > 0 \land \forall B \subseteq A [ B \text{ is measurable } \implies \mu (A)=\mu (B) \lor B \text{ is null } ] \]
\end{definition}




\chapter{Measure-like functions}



In our construction of the Lebesgue measure, we have already encountered various functions that were used as stepping stones to obtain an actual measure (Lebesgue inner measure and length function); many measure-like functions are made with this intention.

Some measure-like functions are made to make functions similar to measures but with more sophisticated codomains. For example, signed measures allow sets with negative measure and vector measures allow the measure of sets to have a direction. Such measures arise in the contextx of mathematical analysis and probability theory, hence it is worth defining them.

\section{Inner and outer measures}
- inner measure
- outer measure

\section{Premeasures}



\section{Signed measures}
Hahn decomposition theorem
Jordan decomposition

\section{Inner and outer measures}
