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\begin{document}

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\section*{measurable subset}

\hypertarget{measurable_subsets}{}\section*{{Measurable subsets}}\label{measurable_subsets}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{modulo_null_sets}{Modulo null sets}\dotfill \pageref*{modulo_null_sets} \linebreak
\noindent\hyperlink{abstract_measurable_sets}{Abstract measurable sets}\dotfill \pageref*{abstract_measurable_sets} \linebreak


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

The measurable subsets of a [[measure space]] $(X,\mu)$ are those [[subsets]] $A$ of the [[underlying set]] $X$ for which the measure $\mu(A)$ is defined (at all, even possibly as infinite). Intuitively, one might expect every subset of $X$ to be measurable, and this is the case in some examples, but in the standard example of [[Lebesgue measure]] on the [[real line]], this is incompatible with the [[axiom of choice]]. (On the other hand, in [[dream mathematics]], where the full axiom of choice fails, every subset of the real line \emph{is} Lebesgue measurable.) Regardless, every subset $A$ of $X$ has both an [[outer measure]] $\mu^*(A)$ and an [[inner measure]] $\mu_*(A)$.

The concept actually makes sense in any [[measurable space]] as well as in related contexts such as [[Cheng spaces]] and [[measurable locales]].

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

Typically, the notion of measurable subset of $X$ is given axiomatically by a [[structure]] on the [[set]] $X$, usually a $\sigma$-[[sigma-algebra|algebra]] $\mathcal{M}$. The a subset of $X$ is \textbf{measurable} if it belongs to $\mathcal{M}$.

Sometimes $\mathcal{M}$ is a weaker structure, such as a $\delta$-ring; see other variants at [[sigma-algebra]]. Then we require some subsidiary notions: $S$ is \textbf{relatively measurable} if $S \cap T$ belongs to $\mathcal{M}$ whenever $T$ does, and $S$ is \textbf{$\sigma$-measurable} if it is a [[union]] of a [[countable set|countable]] [[family of subsets|family]] of elements of $\mathcal{M}$.

(The terms `relatively measurable' and `$\sigma$-measurable' are [[Toby Bartels|my own]]; I cannot find them in the literature. In the case of $\sigma$-measurable sets, the terminology follows a standard pattern. Halmos uses relatively measurable sets, but he doesn't seem to give them a name.)

\hypertarget{modulo_null_sets}{}\subsection*{{Modulo null sets}}\label{modulo_null_sets}

Besides the $\sigma$-algebra of measurable subsets, we may place another structure on $X$, a $\sigma$-[[sigma-ideal|ideal]] $\mathcal{N}$ in $\mathcal{M}$. (This structure also exists, for example, for any [[measure space]], and already for a [[Cheng space]] or a [[localisable measurable space]].) Then a \textbf{[[null set]]} is any subset of $X$ (measurable or not) contained in an element of $\mathcal{N}$. The null sets form a $\sigma$-ideal $\bar{\mathcal{N}}$ of the [[power set]] $\mathcal{P}X$, and we may equivalently begin with the $\sigma$-ideal of null sets as long as every null set is contained in a measurable null set.

This allows two complementary modifications to the notion of measurable set:

\begin{enumerate}%
\item We may accept the [[union]] of any measurable set and any null set as measurable. Since this changes the meaning of `measurable', we may speak of $\mathcal{M}$-measurable and $(\mathcal{M},\mathcal{N})$-measurable sets. The collection of such sets may be denoted $\mathcal{M} \cup \bar{\mathcal{N}}$ (applying $\cup$ pointwise).


\item We may regard two measurable sets as equivalent if their [[symmetric difference]] is a null set (and hence an element of $\mathcal{N}$). This defines an [[equivalence relation]] on $\mathcal{M}$; the collection of [[equivalence classes]] is denoted $\mathcal{M}/\mathcal{N}$.



\end{enumerate}
Note that $(\mathcal{M} \cup \bar{\mathcal{N}})/\mathcal{N} \cong \mathcal{M}/\mathcal{N}$; indeed, this diagram commutes:

\begin{displaymath}
\array {
   \mathcal{M}             & \hookrightarrow & \mathcal{M} \cup \bar{\mathcal{N}} \\
   \downarrow              &                 & \downarrow \\
   \mathcal{M}/\mathcal{N} & \cong           & (\mathcal{M} \cup \bar{\mathcal{N}})/\mathcal{N}
}
\end{displaymath}
Accordingly, one may skip the former modification if one intends to also perform the latter. Nevertheless, even when using $\mathcal{M}/\mathcal{N}$ as the lattice of measurable `sets', if one considers a subset $A$ of $X$ and asks whether $A$ is `measurable', one usually means whether $A \in \mathcal{M} \cup \bar{\mathcal{N}}$.

One could equally well begin with a $\delta$-[[delta-filter|filter]] $\mathcal{F}$, although a $\sigma$-ideal is more traditional. Then a \textbf{[[full set]]} is any subset of $X$ that contains in an element of $\mathcal{F}$. (If we start with the $\delta$-filter $\bar{\mathcal{F}}$ in $\mathcal{P}X$, then every full set must contain a measurable full set.) In [[constructive mathematics]], full sets are more fundamental for such examples as [[Lebesgue measure]]. In any case, the modifications are as follows:

\begin{enumerate}%
\item We may accept the [[intersection]] of any measurable set and any full set as measurable; the collection of such sets may be denoted $\mathcal{M} \cap \bar{\mathcal{F}}$.


\item We may regard two measurable sets as equivalent if their [[biconditional]] is a full set; the collection of equivalence classes is denoted $\mathcal{M}/\mathcal{F}$.



\end{enumerate}
Then we have this commuting diagram:

\begin{displaymath}
\array {
   \mathcal{M}             & \hookrightarrow & \mathcal{M} \cap \bar{\mathcal{F}} \\
   \downarrow              &                 & \downarrow \\
   \mathcal{M}/\mathcal{F} & \cong           & (\mathcal{M} \cap \bar{\mathcal{F}})/\mathcal{F}
}
\end{displaymath}
\hypertarget{abstract_measurable_sets}{}\subsection*{{Abstract measurable sets}}\label{abstract_measurable_sets}

Already we have seen that we may be more interested in [[equivalence classes]] of measurable sets than in the sets themselves. We may well start with any appropriate algebra in the place of $\mathcal{M}/\mathcal{F}$ above and regard its elements are `measurable sets'. This may be done in the theory of [[measurable locales]] and other `pointless' approaches to measure theory.

[[!redirects measurable set]] [[!redirects measurable sets]] [[!redirects measurable subset]] [[!redirects measurable subsets]] [[!redirects measurable subspace]] [[!redirects measurable subspaces]]

[[!redirects relatively measurable subset]] [[!redirects relatively measurable subsets]] [[!redirects relatively measurable set]] [[!redirects relatively measurable sets]]

[[!redirects sigma-measurable subset]] [[!redirects sigma-measurable subsets]] [[!redirects ∞-measurable subset]] [[!redirects ∞-measurable subsets]] [[!redirects sigma-measurable set]] [[!redirects sigma-measurable sets]] [[!redirects ∞-measurable set]] [[!redirects ∞-measurable sets]]



\end{document}