In Measure for Measure, how, and by what means, does our attitude to one of the following change in the course of the play: Angelo, Isabella, the Duke? Discuss the role of Lucio in the play.

1) In answer to this question, I will focus on the Duke. First, I recommend providing a brief background about the Duke. In the play, Duke Vincentio is the Duke of Vienna. Long disturbed by the sinful elements in his beloved city, the Duke thinks that it's high time the city gets cleaned up (morally, that is). So, he tasks Lord Angelo with enforcing all the laws of Vienna, while he supposedly leaves town.
By now, audiences are probably not very impressed with Duke Vincentio. He seems to be abdicating his responsibilities here. After all, as the embodiment of civic power and moral authority, shouldn't he be the one to enforce the laws? However, as the play progresses, Shakespeare presents Duke Vincentio as an altogether different personality. The Duke doesn't really leave town; instead, he disguises himself as Friar Lodowick, and he uses his disguise to engage in some hefty spying work. Additionally, he's not just a strict moralist; he genuinely wants to help people live moral lives (unlike Duke Angelo, who is really a hypocrite in practice).
Here, it would be a great idea to describe how Shakespeare progressively portrays Duke Vincentio, aka Friar Lodowick, as a compassionate authority figure. First, he visits the pregnant Juliet in prison and engages in conversation with her about her "sin." He asks whether Juliet had welcomed Claudio's sexual advances and whether she now repents of the premarital sex. Then, he visits Claudio and later listens in on the conversation between Claudio and Isabella.
During the conversation, both brother and sister are at an impasse. Claudio doesn't want to die, but Isabella doesn't want to give up her virginity either. So, the Duke (Friar Lodowick) proposes that Isabella tell Angelo that she will consent to sleep with him. However, in a twist, Mariana will take Isabella's place during her sexual assignation with Angelo. Here, it would be a great idea to state that Mariana was once betrothed to Angelo himself (Angelo cruelly set Mariana aside when she lost her dowry after her brother's ship sank at sea). So, the Duke imagines that, when Angelo discovers he has slept with Mariana, he will be forced to marry her and to let Claudio go.
Basically, provide examples for how Duke Vincentio displays his compassion and wisdom as the play progresses. By comparing his initial actions with what he manages to accomplish as the play progresses, you will be able to note how your attitudes towards Duke Vincentio change in the course of the play.
2) Lucio is the initial liaison between Claudio and Isabella. In the play, Claudio has been sentenced to death for engaging in premarital sex with Juliet and impregnating her. He begs Lucio to find Isabella and to ask her to intercede on his behalf before Lord Angelo. Lucio does speak to Isabella and after flattering her that men have a soft spot for virgins (Isabella is a novice nun), Isabella promises to speak to Lord Angelo. At this point, Lucio seems a sympathetic and likable character. He even stands in the shadows to "advise" Isabella, as she argues Claudio's case before Angelo.
Despite Lucio's tendency to frequent brothels and to cavort with prostitutes, Lucio comes across as a loyal friend. Through Lucio, Shakespeare portrays the dissolution of the youth of Vienna, but he is also careful to clothe the character with some semblance of humanity. Later in the play, however, Lucio's role serves a different purpose. He is seen to insult the Duke during his conversations with Friar Lodowick (who is, of course, the Duke himself).
Lucio tells Friar Lodowick that the Duke basically indulges in debauchery in private (that he gets drunk on occasion and loves his women). He announces that the Duke is actually "A very superficial, ignorant, unweighing fellow" (Act 3, Scene 2). You can find Lucio's conversation with Friar Lodowick in Act 3, Scene 2. In Act 5, Scene 1, Duke Vincentio (Friar Lodowick) accuses Lucio of calling him a "fool, a coward, One all of luxury, an ass, a madman..." To slander a sovereign or authority figure in Shakespeare's time was considered a very grave offense, one worthy of punishment. In this play, Lucio is made an example of by Shakespeare. That's one of Lucio's main roles in the play.
First, the Duke condemns Lucio to be whipped and then hanged for his slander. Later, however, when it is determined that he has jilted Kate Keepdown, Lucio is ordered by the Duke to marry her. Lucio asserts that being forced to marry Kate is like being subjected to whipping and hanging ("...marrying a punk, my lord, is pressing to death, whipping, and hanging..."). However, the Duke proclaims that "slandering a prince deserves it." So, Lucio has to fulfill his obligations to Kate, however much he disagrees with the "punishment."
Source: Measure for Measure and Lucio, William W. Lawrence, Shakespeare Quarterly, Vol. 9, No. 4 (Autumn, 1958), pp. 443-453