illusion

be under the illusion that wrongly believe that.

2012Daily Telegraph If Barack Obama was under the illusion he didn't have a fight on his hands to win the 2012 US election then he received a rude awakening last night.

be under no illusion (or illusions) be fully aware of the true state of affairs.

1992Christian Scientist Monitor It is crucial to the nation's security…that we be under no illusions about reasons for this zero-loss rate.