\part{Fundamentals}

\chapter{The Lebesgue measure}

We will introduce the notion of measures; typ of set function that assigns sets a nonnegative real number, usually representing some method of "measuring" the size of a set.

Measure theory interacts deeply with general topology, since measure theory is often used as a tool in mathematical analysis.

Before studying the theory generally, it is worth focusing on a specific  measure first; the \emph{Lebesgue measure}.




\section{Lebesgue measure}



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 $2^{\aleph_0}$, 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.



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

\begin{definition}['Ideal' Lebesgue measure]
Let an ideal Lebesgue measure be defined as a set function $\lambda : \mathcal{P}(\mathbb{R}) \to [0,\infty)\cup\{\infty\}$ with the following properties.
\[\lambda([a,b])=b-a\]
\[\lambda(\{x\})=0\]
\[\lambda(A)= \lambda ( \{ x+a \in \mathbb{R} : a \in A \} )\]
\[E_n \text{ are disjoint} \implies \lambda ( \bigcup^{\infty}_{n=0} E_n ) = \sum^{\infty}_{n=0} \lambda (E_n ) \]
\[\]
\end{definition}

If we assume such a set function on $\mathcal{P}(\mathbb{R})$ exists, there would be sets that can prove contradictrions. This means that we need to weaken our definition, or drop the axiom of choice (it turns out that these contradictory sets can only be constructed if one assumes the axiom of choice). The most sensible way to limit our definition is to restrict the Lebesgue measure to sets that won't cause these contradictions.




\subsection{Vitali sets}


We will now construct a notorious set that causes a contradiction with the ideal Lebesgue measure.

\begin{definition}[Vitali set]
\[V \subset [0,1] \text{ is a Vitali set} \iff \forall x \in \mathbb{R} [ \exists ! v \in V [v-x \in \mathbb{Q} ] ] \]
\end{definition}

It is routine to check that Vitali sets are actually sets, and if we assume the axiom of choice (which is standard, since generally mathematicians assume what is called the ZFC set theory), this can indeed be seen since $x \sim y \iff x-y \in \mathbb{Q} \}$ is an equivalence relation over $\mathbb{R}$, hence forms an equivalence class over $\mathbb{R}$ (this step implicitly assumes the axiom of choice). Every equivalence class $[x]$ has some element in $[0,1]$ due to the density of $\mathbb{Q}$ in $\mathbb{R}$, hence Vitali sets are indeed sets.

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.







\subsubsection{Precursors}


Aware that the Lebesgue measure cannot measure all subsets of $\mathbb{R}$, we're prompted to give a constructive definition of the Lebesgue measure and classify what sets the Lebesgue measure can be defined on.

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}





This set function is well defined since $[0,\infty)$ is bounded below and therefore due to the completeness property of $\mathbb{R}$, an infimum will always exist.

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 for all subsets of $\mathbb{R}$, since that would make $\lambda^{*}$ exactly obey those 'idealized' Lebesgue measure properties, but then the Vitali sets can be used as a contradiction.


The benefit of the Lebesgue outer measure is that it is \emph{extremely} close to what we want from the Lebesgue measure; even the value it produces is what we want! The only caveat is that it is only countably subadditive in general; to promote it to the Lebesgue measure, we restrict it to the sets on which it happens to be countably additive

We desire our Lebesgue measure to have countable additivity for disjoint sequence of subsets, so to see what sets the Lebesgue measure works on we restrict the domain of $\lambda^{*}$ from $\mathcal{P}(\mathbb{R})$ to $\Sigma =\{ \bigcap^{\infty}_{n=1} E_n \in \mathcal{P}(\mathbb{R}) : (E_n)_{n \in \mathbb{N}} \text{ are pairwise disjoint } \implies  \lambda^{*}( \bigcup^{\infty}_{n=1}E_n) = \sum^{\infty}_{n=1} \lambda^{*}( E_n) \}$.

These will be the sets for which our Lebesgue measure will work on.

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

	Let $\mathcal{L} \subset \mathbb{R}$ represent the Lebesgue measurable sets.
\end{definition}

An equivalent (and much more compact) way to characterize this is through \emph{Carathéodory criterion}.

\begin{theorem}[Carathéodory criterion]
A subset $E$ of $\mathbb{R}$ is Lebesgue measurable if the following holds.
\[\lambda^{*}(A) = \lambda^{*}(A\cap E) + \lambda^{*}(A \cap (\mathbb{R} \setminus E))\]
\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{proposition}
\[\lambda^{*}(E)=0 \implies E \text{ is Lebesgue measurable}\]
Let $I \subseteq \mathbb{R}$ be an interval, then $I$ is Lebesgue measurable.
Open sets in the Euclidean topology on $\mathbb{R}$ are Lebesgue measurable.
\end{proposition}

One can prove the following using set theory and the fact that the emptyset obeys $\lambda^{*}(\emptyset)=0$.

\begin{proposition}
	$\mathcal{L}$ is closed under countable intersections
	$\mathcal{L}$ is closed under complement relative to $\mathbb{R}$
	$\mathbb{R}\in \mathcal{L}$ 
\end{proposition}


These laws show an "algebraic" nature of the Lebesgue measurable sets, and indeed, we will later see that these laws are so important that all other measures obey these laws.

Since open intervals are Lebesgue measurable and open intervals are a topological basis for $\mathbb{R}$ (i.e any open set is a countable union of some open intervals), we have the following.


\begin{corollary}
\end{corollary}



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



\subsection{Lebesgue null sets}

\begin{definition}[Lebesgue null set]
\[E \text{ is a }\lambda-\text{null set} \iff \lambda (E) = 0\]
\end{definition}


One can trivially see that finite sets are always Lebesgue null sets (calculate the Lebesgue measure by the disjoint union of its singletons), however many infinite sets are also Lebesgue null sets.

\begin{proposition}
Countable subsets of $\mathbb{R}$ are Lebesgue null sets.
\end{proposition}

There are also uncountable Lebesgue null sets out there.

\begin{proposition}
The Cantor set is a Lebesgue null set.
\end{proposition}
























































































\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.



\section{$\sigma$-algebrae}


Now that we have developed and studied the Lebesgue measure, the goal of this chapter is to develop a definition of a "measure" in general. We will start by deciding what intutional properties we desire a generic measure on a space $X$ could have (without assuming any properties of the space $X$ to keep our measure as general as possible).


\begin{definition}[Measure]
A \emph{measure} is a set function $\mu : \Sigma \subseteq \mathcal{P}(X) \to [0,\infty)\cap\{\infty\}$ with the following properties.
\begin{itemize}
	\item $\emptyset \in \Sigma \land  \mu(\emptyset) = 0 $
	\item $ E_n \in \Sigma \implies \mu ( \bigcup^{\infty}_{n=1} E_n ) =  \sum^{\infty}_{n=1}  \mu ( E_n )$
	\item $E \in \Sigma \implies \mu ( X ) =  \mu(X \setminus E) + \mu(E)$
\end{itemize}
\end{definition}

The first condition obviously means $\emptyset$ should be in $\Sigma$ since it assigns $\emptyset$ a value, specifically 0.

Countable additivity means that any countable union of measurable sets should be in $\Sigma$, since the condition implies a way to calculate the measure of such unions.

The final condition gives a way to measure the space $X$ itself, hence it should be in $\Sigma$. Furthermore the same condition also implies that the complement of any measurable set is in $\Sigma$.

Recall that we had shown the set of Lebesgue measurable sets to obey these "algebraic" laws, and now we have discovered that they are consisten with our idea of general measures too. 


A family of sets following these laws is known as a \emph{$\sigma$-algebra}. 

\begin{definition}[$\sigma$-algebra]
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{Measurable space}

Once a $\sigma$-algebra $\Sigma$ has been chosen for a space, we can form a \emph{measurable space}; a space with the potential to have a measure on $\Sigma$ constructed.


\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}

There is much we can say about these spaces, even though we haven't even chosen our

\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 $X$




\section{Measure spaces}
Measurable spaces obviously have potential for a measure to be endowned onto it, while a \emph{measure space} is a measurable space with a specific measure attached.

We can equivalently reword our definition of measures in terms of $\sigma$-algebrae, just to make things cleaner.

\begin{definition}[Equivalent definition of a 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}[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 $\mu : \Sigma \to [0,\infty)\cup\{\infty\}$ 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.


Though generalizing measurable sets requires only a measurable space, the notion of null sets is dependent on a measure and hence only exists in measure spaces.

\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}


\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}


\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}



Note that the entire last chapter was essentially the study of a particulr measure space, called the \emph{Lebesgue measure space}.

\begin{definition}[Lebesgue measure space]
\[(\mathbb{R}, \mathcal{L} , \lambda)\]
\begin{itemize}
	\item $\mathbb{R}$ is the set of real numbers
	\item $\mathcal{L} = \{ E \subseteq \mathbb{R} : \forall A \subseteq \mathbb{R} [  \lambda^{*}(A) = \lambda^{*}(E \cap A) + \lambda^{*} ([\mathbb{R}\setminus E] \cap A) ] \}$ is the $\sigma$-algebra
	\item $\lambda : \mathcal{L} \to [0,\infty)\cup\{\infty\}$ where $\lambda(E) = \lambda^{*}(E)$ is the Lebesgue measure
\end{itemize}
\end{definition}

Note that $\mathcal{B}(\mathbb{R},\mathcal{T}_{\mathbb{R}}) \subset \mathcal{L}$.


\section{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.

\subsection{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.

\begin{definition}[Finite measure space]
A \emph{finite measure space} $(X,\Sigma , \mu)$ is a measure space such that $\mu (E) < \infty$ for any $\Sigma$-measurable $E$.
\[ (X,\Sigma,\mu) \text{ is finite } \iff \forall E \in \Sigma [ \mu (E) < \infty ] \]
\end{definition}

\subsection{Complete measure}

\begin{definition}[Complete measure space]
	\[ (X,\Sigma,\mu) \text{ is complete } \iff [ \mu (E) = 0 \land F \subseteq E \implies F \in \Sigma ] \]
\end{definition}



\subsection{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}




\section{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); one can indeed consider measure-like functions for more specifc use cases, or as an intermediate step for defining an actual measure (like what we did for the Lebesgue measure).

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.

\subsection{Inner and outer measures}




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

Outer measures are very useful for designing actual measures since they can measure any powerset, and the actual measure function is typically defined by restricting this outer measure's domain to sets satisfying a Caratheorody criterion.

\subsection{Premeasures}



\subsection{Signed measures}


Measure that allows Negative values


\begin{definition}[Signed measure]
	A \emph{signed measure on $(X,\Sigma)$} is a function $\mu : \Sigma \to \mathbb{R}\cup\{\pm\infty\}$ 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}

As it turns out, when a signed measure is present on a measurable space one can partition said space into a 'positive' and 'negative' region ($P,N$ respectively) on which the signed measure is always positive on a subset of the positive space and negative on a subset of the negative space.

The thought that inspires this theorem is by considering a measurable set such that all its measurable subsets have nonnegative measure, finding a way to inductively grow this set while retaining this property (calling the limit of this as $P$), and considering $N=X\setminus P$.



\begin{theorem}[Hahn decomposition theorem]
Given a measurable space $(X,\Sigma)$ and signed measure $\mu : \Sigma \to \mathbb{R}\cup\{\pm \infty\}$  there exists a pair of $\Sigma$-measurable sets $(P,N)$ called a \emph{Hahn decomposition} such that:
\begin{itemize}
	\item $P,N$ partition $X$
	\item $E \subseteq P \implies \mu(E) \geq 0$
	\item $E \subseteq N \implies \mu(E) \leq 0$
\end{itemize}
Moreover, this Hahn decomposition is essentially unique (the symmetric difference between all other Hahn decompositions are $\mu$-null sets)
\end{theorem}

The existence of this space decomposition allows for a measure decomposition too; a signed measure can be represented by an expression of 2 actual measures.

\begin{corollary}[Jordan decomposition theorem]
	Given a measurable space $(X,\Sigma)$ and signed measure $\mu : \Sigma \to \mathbb{R}\cup\{\pm \infty\}$  there exists a pair of $\Sigma$-measures $(\mu^{+},\mu^{-})$ called a \emph{Jordan decomposition} such that:
\begin{itemize}
	\item $\mu(E) = \mu^{+}(E) - \mu^{-}(E)$
	\item At least one of these measures is finite
	\item For any Hahn decomposition $(P,N)$, $\mu^{+}(E) = \mu(E \cap P)$
	\item For any Hahn decomposition $(P,N)$, $\mu^{-}(E) = -\mu(E \cap N)$
\end{itemize}
Moreover, this Jordan decomposition is unique.
\end{corollary}


When developing the Lebesgue integral, we will initially be restricted to nonnegative functions, however the Jordan decomposition theorem will allow us to extend Lebesgue integration to drop this requirement.











\section{Borel spaces}



It was mentioned that topology and measure theory often cross paths; this section will motivate .

Given a topological space $(X,\mathcal{T})$, imagine we want to find a measurable space $(X,\Sigma)$ with potential to measure all open sets (so we want $\mathcal{T} \subseteq \Sigma$).

The minimal possible $\sigma$-algebra would be the one obtained by taking the topology $\mathcal{T}$ and "closing it up" under the $\sigma$-algebra laws (transfinite-inductively taking complements and countable intersections) to give us our $\Sigma$; this is known as the \emph{Borel $\sigma$-algebra of $(X,\mathcal{T})$}, and the measurable space $(X,\Sigma)$ is the \emph{Borel space of $(X,\mathcal{T})$}.




\begin{definition}[Borel $\sigma$-algebra]
Given a topological space $(X,\mathcal{T})$, the \emph{Borel $\sigma$-algebra of $(X,\mathcal{T})$} is the minimal $\sigma$-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}

We then define the Borel space as the measurable space of the Borel $\sigma$-algebra.

\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}


Since $\sigma$-algebrae are by definition closed under complement, we immediately obtain the following.

\begin{proposition}
Closed sets of $(X,\mathcal{T})$ are Borel sets of $\mathcal{B}(X,\mathcal{T})$.
\end{proposition}



\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}


\subsection{Euclidean Borel sets versus Lebesgue measurable sets}


We end the chapter by comparing the Borel $\sigma$-algebra of the Euclidean topological space of $\mathbb{R}$

\begin{theorem}
$\mathcal{B}(X,\mathcal{T}_{\mathbb{R}}) \subset \mathcal{L}$
\end{theorem}


Since the intervals are by definition a basis for the Euclidean topology (i.e open sets are defined the countable unions of intervals), and the Lebesgue measure works on all intervals as well as countable unions of them, all open sets of the Euclidean topology on $\mathbb{R}$ are clearly also Lebesgue measurable sets.

What is nowhere near as clear is that $\mathcal{L}$ actually includes a non-Borel set of the Euclidean topology on $\mathbb{R}$! These sets are rather weird; here's an example of by Nikolai Luzin, 

\begin{definition}[Luzin's non-Borel set]
\[A = \{ x \in \mathbb{R} : x = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \ddots}}} \land \exists \{a_{n_k}\} [ a_{n_k} | a_{n_{k+1}} ]  \}  \]
\end{definition}