Structure of level sets: inclusion

 
 

Monotonicity for inclusion.

Decreasingness of superior level sets:
$\forall\,\lambda \leq \mu,\quad [u\geq \lambda] \supseteq [u\geq \mu]$

Increasingness of inferior level sets:
$\forall\,\lambda \leq \mu,\quad [u<\lambda] \subseteq [u< \mu]$

Summer School on Mathematical Problems in Image Processing, Trieste, Italy, September 4th-18th 2000