ThmDex – An index of mathematical definitions, results, and conjectures.

Indicator function under scaling of the argument

Formulation 0

Let $\mathbb{R}^N$ be a D816: Euclidean real Cartesian product such that

(i)	$E \subseteq \mathbb{R}^N$
(ii)	$I_E$ is an D41: Indicator function on $\mathbb{R}^N$ with respect to $E$
(iii)	$x \in \mathbb{R}^N$

Let $a \in \mathbb{R}$ be a D993: Real number such that

(i)	\begin{equation} a \neq 0 \end{equation}

Then

(1)	\begin{equation} I_E(a x) = I_{a^{-1} E} (x) \end{equation}
(2)	\begin{equation} I_E (a^{-1} x) = I_{a E} (x) \end{equation}

Formulation 1

Let $\mathbb{R}^N$ be a D816: Euclidean real Cartesian product such that

(i)	$E \subseteq \mathbb{R}^N$
(ii)	$I_E$ is an D41: Indicator function on $\mathbb{R}^N$ with respect to $E$
(iii)	$x \in \mathbb{R}^N$

Let $a \in \mathbb{R}$ be a D993: Real number such that

(i)	\begin{equation} a \neq 0 \end{equation}

Then

(1)	\begin{equation} I_E(a x) = I_{E / a} (x) \end{equation}
(2)	\begin{equation} I_E (x / a) = I_{a E} (x) \end{equation}

Formulation 2

Let $\mathbb{R}^N$ be a D816: Euclidean real Cartesian product.
Let $I_E$ be an D41: Indicator function for $E \subseteq \mathbb{R}^N$ in $\mathbb{R}^N$.
Let $a, x \in \mathbb{R}$ each be a D993: Real number such that

(i)	$a \neq 0$

Then

(1)	\begin{equation} I_E(a x) = I_{\frac{1}{a} E} (x) \end{equation}
(2)	\begin{equation} \textstyle I_E (\frac{1}{a} x) = I_{a E} (x) \end{equation}

Proofs

Proof 0

Let $\mathbb{R}^N$ be a D816: Euclidean real Cartesian product such that

(i)	$E \subseteq \mathbb{R}^N$
(ii)	$I_E$ is an D41: Indicator function on $\mathbb{R}^N$ with respect to $E$
(iii)	$x \in \mathbb{R}^N$

Let $a \in \mathbb{R}$ be a D993: Real number such that

(i)	\begin{equation} a \neq 0 \end{equation}

Using result R4587: and the definition of an indicator function, we have the chain of equivalencies \begin{equation} \begin{split} I_E(a x) = 1 \quad & \iff \quad a x \in E \\ & \iff \quad x \in a^{-1} E \\ & \iff \quad I_{a^{-1} E}(x) = 1 \\ \end{split} \end{equation} whence the first claim follows as a consequence of R2965: Indicator function is uniquely identified by its support. The second claim follows by substituting $a^{-1} \neq 0$ in the place of $a$. $\square$