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

Result R2182 on D2948: Sublevel set

Isotonicity of real sublevel sets

Formulation 0

Let $f : X \to \mathbb{R}$ be a D4364: Real function.
Let $a, b \in \mathbb{R}$ each be a D993: Real number such that

(i)	$a \leq b$

Then \begin{equation} \{ f \leq a \} \subseteq \{ f \leq b \} \end{equation}

Formulation 1

Let $f : X \to \mathbb{R}$ be a D4364: Real function.
Let $a, b \in \mathbb{R}$ each be a D993: Real number such that

(i)	$a \leq b$

Then \begin{equation} \{ x \in X : f(x) \leq a \} \subseteq \{ x \in X : f(x) \leq b \} \end{equation}

Subresults

▶	R4182: Isotonicity of sublevel probabilities for a random real number

Proofs

Proof 0

Let $f : X \to \mathbb{R}$ be a D4364: Real function.
Let $a, b \in \mathbb{R}$ each be a D993: Real number such that

(i)	$a \leq b$

If $x \in \{ f \leq a \}$, then $f(x) \leq a \leq b$, so that $x \in \{ f \leq b \}$. Since $x \in \{ f \leq a \}$ was arbitrary, we have the inclusion $\{ f \leq a \} \subseteq \{ f \leq b \}$. $\square$