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\begin{document}

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\section*{simple function}

\hypertarget{simple_functions}{}\section*{{Simple functions}}\label{simple_functions}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{integration}{Integration}\dotfill \pageref*{integration} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Simple functions are (almost) the most basic notion of [[measurable function]] in [[measure theory]]. Given a [[measure]], it's easy to define the [[integral]] of a simple function, and we extend this to more general functions by continuity.

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

Let $X$ be a [[measurable space]]. We may want $X$ to be equipped with some more data; if $X$ is a [[measure space]], then this is plenty of data. However, for the most basic definitions, it's enough if $X$ is simply a measurable space. This is the [[domain]] of our simple functions.

Another necessary datum is the simple functions' [[codomain]] $K$, which we will eventually want to be at least a [[Banach space]] over the [[real numbers]]. (In the simplest example, $K$ is $\mathbb{R}$ itself, or perhaps the space $\mathbb{C}$ of [[complex numbers]].) We take $K$ to be a measurable space using its [[Borel sets]].

\begin{defn}
\label{trad}\hypertarget{trad}{}
A [[measurable function]] from $X$ to $K$ is \textbf{simple} if its [[range]] is [[finite set|finite]].

\end{defn}
Since a simple function $f$ is measurable and a [[singleton subset|singleton]] is Borel, each [[fibre]] of $f$ is a [[measurable set]] in $X$; the function $f$ is given by the (finitely many) nonempty fibres and their (singleton) images. This suggests another way to look at simple functions:

\begin{defn}
\label{lincomb}\hypertarget{lincomb}{}
A \textbf{simple function} from $X$ to $K$ is a formal $K$-[[linear combination]] of [[measurable subsets]] of $X$.

\end{defn}
Here we identify a measurable set $A$ with its [[characteristic function]] $\chi_A$, so the formal linear combination $\sum_i c_i A_i$ is identified with the function $\sum_i c_i \chi_{A_i}$, which is measurable and whose range is contained in the finite set of sums of the $c_i$. (If there are $n$ terms in the linear combination, then there are at most $2^n$ such sums.)

However, the na\"i{}ve notion of equality of linear combinations is finer than equality of the corresponding functions, so we must combine Definition \ref{lincomb} with a definition of equality:

\begin{defn}
\label{equal}\hypertarget{equal}{}
Two simple functions from $X$ to $K$, in the sense of Definition \ref{lincomb}, are \textbf{equal} if their corresponding [[functions]] from $X$ to $K$ are equal as functions.

\end{defn}
Then we have a [[canonical transformation|canonical]] [[bijection]] between the set of simple functions as in Definition \ref{trad} and the set of [[equivalence classes]] of simple functions as in Definition \ref{lincomb}.

Arguably, even this is not really the correct notion of equality, since functions may be equal for the purpose of integration without being literally equal. If $X$ is equipped with a $\sigma$-[[sigma-ideal|ideal]] of [[null sets]] (or a $\delta$-[[delta-filter|filter]] of [[full sets]]), then we may consider a yet coarser notion of equality:

\begin{defn}
\label{almostequal}\hypertarget{almostequal}{}
Two simple functions, in the sense of either Definition \ref{trad} or Definition \ref{lincomb}, are \textbf{almost equal} if they (or their corresponding functions) are [[almost equality|equal almost everywhere]].

\end{defn}
Sometimes, we wish to restrict attention to those simple functions which we expect to have a finite integral. If $X$ is equipped with an [[ideal]] of [[bounded sets]] (which in a measure space are sets with finite measure), then we may do this:

\begin{defn}
\label{bounded}\hypertarget{bounded}{}
A \textbf{simple function of bounded support} is a simple function in the sense of Definition \ref{trad} such that the fibre over every non-zero number is bounded, or equivalently (in the sense of Definition \ref{lincomb}) a formal linear combination of bounded measurable sets.

\end{defn}
In some approaches to [[measure theory]], one \emph{starts} with a $\delta$-[[delta-ring|ring]] of measurable sets, which may be reinterpreted as the bounded sets in the generated $\sigma$-[[sigma-algebra|algebra]] of [[relatively measurable sets]], and then the simple functions will automatically have bounded support.

Finally, there is one more useful restriction (and slight generalisation) of simple functions, applicable when $K$ is ordered:

\begin{defn}
\label{positive}\hypertarget{positive}{}
A \textbf{positive simple function} is a simple function in the sense of Definition \ref{trad} whose range is contained in the [[positive cone]] $K^+$ of $K$, or equivalently (in the sense of Definition \ref{lincomb}) a formal $K^+$-linear combination of measurable sets. An \textbf{extended positive simple function} (note the [[red herring]]) takes values in the [[extended positive cone]] $\bar{K}^+$, or equivalently is a $\bar{K}^+$-linear combination.

\end{defn}
\hypertarget{integration}{}\subsection*{{Integration}}\label{integration}

Let $X$ be equipped with a [[measure]] $\mu$, so $(X,\mu)$ is a [[measure space]]. (In particular, $X$ has the structure necessary for all of the definitions above, including both Definitions \ref{almostequal} and \ref{bounded}.)

If $f$ is a simple function from $X$ to $K$, then we wish to define the [[integral]] of $f$. In general, this is a little tricky, but it's easy if $f$ either is \hyperlink{positive}{positive} or has \hyperlink{bounded}{bounded support}. It is easiest to write down the definition if we think of simple functions using Definition \ref{lincomb}. Then we have:

\begin{defn}
\label{integral}\hypertarget{integral}{}
The \textbf{integral} of the simple function $f$, represented by the linear combination $\sum_i c_i A_i$, is $\sum_i c_i \mu(A_i)$.

\end{defn}
\begin{prop}
\label{intpositive}\hypertarget{intpositive}{}
The integral of a positive simple function always exists (but may be infinite). It is finite if $\mu$ is a [[finite measure]], and it is positive (possibly $0$ or $\infty$) if $\mu$ is a [[positive measure]]. Also, if $\mu$ is positive, then the integral of an extended positive simple function always exists.

\end{prop}
(However, the integral of an extended positive simple function with respect to a finite positive measure need not be finite.)

\begin{prop}
\label{intbounded}\hypertarget{intbounded}{}
The integral of a simple function with bounded support always exists and is finite (being a finite linear combination of finite numbers).

\end{prop}
\begin{prop}
\label{intequal}\hypertarget{intequal}{}
Two (positive or with bounded support) simple functions $f$ and $g$ are almost equal (with respect to $\mu$) if and only if the integral of $f - g$ is zero.

\end{prop}
\begin{defn}
\label{norm}\hypertarget{norm}{}
The \textbf{$L^1$-norm} of a simple function is the integral of its pointwise norm (which is a positive simple function to $\mathbb{R}$) with respect to the [[absolute value]] of the measure $\mu$ (which is a positive measure):

\begin{displaymath}
{\|f\|}_1 \coloneqq \int {\|f(x)\|} {|\mu(\mathrm{d}x)|} .
\end{displaymath}
\end{defn}
In this context, we usually start with a positive measure $\mu$; in that case, of course, there is no need to bother taking the absolute value of $\mu$.

\begin{prop}
\label{NVS}\hypertarget{NVS}{}
The simple functions of bounded support form a [[normed vector space]] $Simp_c$ under the $L^1$-norm, if we consider them up to \hyperlink{almostequal}{almost equality}.

\end{prop}
If we don't use almost equality, then we get in general only a [[seminorm]], but if we pass to a [[quotient space]] with a norm, then Proposition \ref{intequal} tells us that we are now using almost equality (and shows that Definition \ref{integral} is well defined when applied to Definition \ref{trad}).

\begin{defn}
\label{L1}\hypertarget{L1}{}
The [[complete space|completion]] of the normed vector space $Simp_c$ (under the $L^1$-norm) is the [[Banach space]] $L^1$ of \textbf{[[absolutely integrable functions]]} (an example of a [[Lebesgue space]]).

\end{defn}
\begin{prop}
\label{integration}\hypertarget{integration}{}
Taking the integral of a simple function of bounded support is a [[continuous linear functional]] on $Simp_c$, so it extends to all of $L^1$.

\end{prop}
In this way, we may define the integral of any absolutely integrable function.

There might be some technical requirements for this to be true. I'll try to check on that.

[[!redirects simple function]] [[!redirects simple functions]]



\end{document}