Sigh Of Relief Meaning

(idiomatic) A reassurance or support, something that reduces stress from an arduous activity.

Example: A release of stress through breathing motions.
2005, Erich Hartung, America's Finest[1], page 149:
  Boyd was put in charge of SWAT three years later. Indeed, the force of the elite SWAT Team was a big sigh of relief for the other police offices on the ground. However, for the bank robbers, it was not.