Subjects: Measure Theory
Links: Measure Spaces and Measurable Spaces, Rings and Algebras of Sets

Def: Suppose that $(X,\mathcal{S})$ is a measurable space. For every $f:X\to \mathbb{R}$, we shall write $N(f):=f^{-1}[\mathbb{R}\setminus \{0\}]$; if a real valued function such that, for every Borel subset $M$ of the real line the set $N(f)\cap f^{-1}[M]$ is measurable, then $f$ is called a measurable function.

If $f$ is a measurable function on $X$ and if we take $M=\mathbb{R}$, then it follows that $N(f)$ is a measurable set. Hence if $E$ is measurable subset of $X$ and if $M$ is Borel subset of the real line, then it follows that $E\cap f^{-1}[M]$ is measurable.

In other words, if we say that real valued function defined on a measurable set $E$ is to be called a measurable on $E$ whenever $E\cap f^{-1}[M]$ is measurable for every set $M$, then we have proved that a measurable function is measurable on every measurable set. If in particular, the entire space $X$ happens to measurable, measurable function is one whose inverse maps the sets of one prescribed $\sigma$-ring into the sets of another prescribed $\sigma$-ring.

It is clear that the concept of measurabilty for a function depends on the $\sigma$-ring $\mathcal{S}$ and therefore, we shall say that a function is measurable with respect to $\mathcal{S}$, or, more concisely that is measurable ($\mathcal{S}$).

If in particular $X=\mathbb{R}$, and $\mathcal{B}(\mathbb{R})$ and $\overline{\mathcal{B}(\mathbb{R})}$ are the Borel sets and the family if Lebesgue measurable sets respectively, then we shall call a function measurable with respect to $\mathcal{B}(\mathbb{R})$ a Borel measurable function, and a function that is measurable with respect to $\overline{\mathcal{B}(\mathbb{R})}$ a Lebesgue measurable function.

This concept is quite close to topological continuity.

We need to extend the concept of measurability for extended real functions also. We define the concept by making the convention that one-point sets $\{\mathrm{\infty }\}$ and $\{-\mathrm{\infty }\}$ of the extended real line are to be regarded as Borel sets. Accordingly a possibly infinite valued function $f$ is measurable, if, for every Borel set $M$, each of the three sets $$f^{-1}{\infty}, \qquad, f^{-1}{-\infty}, \qquad N(f)\cap f^{-1}[M]$$is measurable.

Th: Let $f$ be a real valued function on a measurable space $(X,\mathcal{S})$. The following statements are equivalent:

• $f$ is measurable.
• For every $c\in \mathbb{R}$, the set $N(f)=\{x\mid f(x)<c\}=N(f)\cap f^{-1}[(-\mathrm{\infty },c)].$
• For every $c\in \mathbb{R}$, the set $N(f)=\{x\mid f(x)\le c\}=N(f)\cap f^{-1}[(-\mathrm{\infty },c]].$
• For every $c\in \mathbb{R}$, the set $N(f)=\{x\mid f(x)>c\}=N(f)\cap f^{-1}[(c,\mathrm{\infty })].$
• For every $c\in \mathbb{R}$, the set $N(f)=\{x\mid f(x)\ge c\}=N(f)\cap f^{-1}[[c,\mathrm{\infty })].$

Cor: Let $f$ be a real valued function on a measurable space $(X,\mathcal{S})$, and $D\subseteq \mathbb{R}$ dense. The following statements are equivalent:

• $f$ is measurable.
• For every $c\in D$, the set $N(f)=\{x\mid f(x)<c\}=N(f)\cap f^{-1}[(-\mathrm{\infty },c)].$
• For every $c\in D$, the set $N(f)=\{x\mid f(x)\le c\}=N(f)\cap f^{-1}[(-\mathrm{\infty },c]].$
• For every $c\in D$, the set $N(f)=\{x\mid f(x)>c\}=N(f)\cap f^{-1}[(c,\mathrm{\infty })].$
• For every $c\in D$, the set $N(f)=\{x\mid f(x)\ge c\}=N(f)\cap f^{-1}[[c,\mathrm{\infty })].$

Prop: If $f$ is a measurable function and $c\in \mathbb{R}$, then $cf$ is measurable.

Prop: Let $E\subseteq X$ and $(X,\mathcal{S})$ is measurable space. $E$ is measurable iff ${\chi }_E$ is measurable.

Prop: A nonzero constant function is measurable iff $X$ is measurable.

Prop: If $f:\mathbb{R}\to \mathbb{R}$ is increasing then $f$ is Borel measurable.

Prop: Let $(X,\tau )$ be a topological space. If $f:X\to \mathbb{R}$ is continuous, then $f$ is Borel measurable.

Prop: Suppose that $f$ is a real valued function on a measurable space $(X,\mathcal{S})$, and for every $t\in \mathbb{R}$, write $B(t)=\{x\mid f(x)\le t\}$. Then

• $s<t$ implies $B(s)\subseteq B(t)$.
• $\bigcup _{t\in \mathbb{R}}B(t)=X$ and $\bigcap _{t\in \mathbb{R}}B(t)=\varnothing$.
• $\bigcap _{s<t}B(t)=B(s)$.
  Conversely, if $\{B(t)\mid t\in \mathbb{R}\}$ with the properties above, then there exists a unique, finite, real valued function $f$ such that $\{x\mid f(x)\le t\}=B(t)$. The unique function is $f(x)=inf\{t\mid x\in B(t)\}$.

Obs: If $f$ is a measurable function on a totally finite measure space $(X,\mathcal{S},\mu )$ and if, for every Borel set $M$ on the extended real line, we write $\nu (M)=\mu (f^{-1}[M])$, then the $\nu$ measure on the family of Borel sets, and it is commonly referred as the pushforward measure or the image measure of $\mu$ under the function $f$. The pushforward measure is often denoted as $f_{\ast }\mu .$

Def: If $f$ is finite valued, then the function $g:\mathbb{R}\to \mathbb{R}$, defined by $g(t)=\mu (\{x\mid f(x)<t\})$, then $g$ is called the distribution of $f$.

Prop: If $f$ is finite valued, then the $g$ is the distribution of $f$, is increasing, continuous on the left, and such that $g(-\mathrm{\infty })=0$ and $g(\mathrm{\infty })=\mu (X)$.

Prop: Let $f$ be finite valued, and $g$ is the distribution of $f$. If $g$ is continuous, then the Lebesgue-Stieltjes measure ${\mu }_g$ is the completion of $\nu$.

Prop: If $E$ is a measurable set and its characteristic function ${\chi }_E$, then $({\chi }_E{)}_{\ast }\mu (M)={\chi }_M(1)\mu (E)+{\chi }_M(0)\mu (X\setminus E)$

Def: A complex valued function $f:X\to \mathbb{C}$ is measurable if $\mathrm{\Re }(f),\mathrm{\Im }(f):X\to \mathbb{R}$ are measurable.

Prop: A complex valued function $f$ is measurable iff for every open set $M$ in the complex plane, the set $N(f)\cap f^{-1}[M]$ is measurable.

Combinations of Measurable Functions

Th: If $f$ and $g$ are extended real measurable functions on a measurable space $(X,¸\mathcal{S})$, and if $c\in \mathbb{R}$, then each of the three sets: $$A = {x\mid f(x) < g(x) + c}, \qquad B = {x\mid f(x) \le g(x) + c}, \qquad C = {x\mid f(x) = g(x) + c},$$has measurable intersection with every measurable set.

Th: If $\varphi$ is an extended real valued Borel measurable function on the extended real line such that $\varphi (0))=0$, and if $f$ is an extended real valued measurable function on a measurable space $X$, then the function $\tilde{f}$, defined by $\tilde{f}(x):=\varphi (f(x))$, is a measurable function on $X$.

Obs: For each $\alpha >0$, the function $\varphi (x)=|x{|}^{\alpha }$, is Borel measurable.

Cor: If $\varphi$ is an extended real valued Borel measurable function on the extended real line, if $f$ is an extended real valued measurable function on a measurable space $X$ with $X$ is measurable, then the function $\tilde{f}$, defined by $\tilde{f}(x):=\varphi (f(x))$, is a measurable function on $X$.

Cor: Let $f$ be an extended real valued measurable function on a measurable space $X$, and $\alpha >0$, then $|f{|}^{\alpha }$ is measurable.

Prop: If $f$ and $g$ are extended real valued measurable functions on a measurable space $X$, then so also are $f+g$ and $fg$.

Cor: If $f$ and $g$ are extended real valued measurable functions on a measurable space $X$, then so also are $min\{f,g\}$ and $max\{f,g\}$.

Def: Let $f$ is an extended real valued measurable function on a measurable space $X$, then we define the functions $f^{+}:=max\{f,0\}$ and $f^{-}:=-min\{f,0\}$ are called the positive part and the negative part of $f$, respectively. Both are measurable.