In Measure for Measure, how is the theme of "arbitary authority" depicted?
Your question touches upon a much debated issue in this excellent play, which is the way in which the play presents us with different kinds of authority and governance and tries to search for a balance between the arbitary authority of Angelo and the ruthlessness with which he pursues justice and the somewhat laxer system of governance endorsed by Duke Vincentio. Let us remember that the Duke himself in Act I admits that he has been too lax in his role of authority, but he also says that he suspects Angelo will leap to the other extreme. Let us also acknowledge the way in which Isabella herself represents another form of arbitrary authority as her flat refusal to compromise on her own standards indicates.
The oscillation between these two extremes, between the arbitary authority of Angelo and the laxer approach to justice demonstrated by Vincenio reaches its climax in Act V when Vincentio himself delivers severe sentences on Angelo and Lucio, only then going on to rescind them moments later. Clearly the play presents arbitrary authority as a bad system of governance, but the ending of the play is problematic in the way that Vincentio himself has shown himself to be rather arbitrary in the way that he hs achieved a "fair"ending. Let us not forget that he has deliberately manipulated and deceived other characters in the name of "fairness," showing himself to be rather arbitrary too.