I want to ask about Hamlet 2.2. How effective is Polonius as a bearer of news? Is Claudius convinced?

In the second scene of Act 2 of Hamlet, Polonius is truly the bearer of news; however, he is most certainly not effective and, in fact, proves himself to be the fool of the play.  Gertrude and Claudius have been waiting impatiently for Polonius to reveal any word about the reason behind Hamlet's madness.  (In the previous act, Claudius and Gertrude send Polonius out to spy on Hamlet.)  When it is the appropriate time for Polonius to share the news, Polonius does nothing but bungle his words:

My liege and madam, to expostulate / What majesty should be, what duty is, / Why day is day, night night, and time is time, / Were nothing but to waste night, day, and time. / Therefore, since brevity is the soul of wit, / And tediousness the limbs and outward flourishes, / I will be brief.  Your noble son is mad. (2.2.86-93)

The irony here is that Polonius is not being brief at all.  By describing how time should never be wasted, Polonius himself wastes time!  Gertrude herself does all but roll her eyes at Polonius while saying, "More matter, with less art." (2.2.95) However, even after being admonished by Gertrude, Polonius still goes on:

Madam, I swear I use no art at all. / That's he's mad, 'tis true: 'tis true 'tis pity, / And pity 'tis 'tis true--a foolish figure.  (2.2.96-98)

A foolish figure, Polonius?  Yes, I would say so, but not about Hamlet.  Even in Polonius' eventual revelation of the actual news (a letter from Hamlet to Ophelia), Polonius still proves his inept ability to read something so simple as a letter:

"To the celestial, and my soul's idol, the most / beautified Ophelia"-- / That's an ill phrase, a vile phrase; "beautified" is a / vile phrase.  But you shall hear.  Thus:  / "In her excellent white bosom, these, &c." (2.2.109-113)

As the King and Queen lean forward, begging to hear the news of Hamlet's madness, Polonius still inserts his own views on Hamlet's flirtations.  This goes on and on (and on) until the three finally come to the conclusion that Hamlet is lovesick for Ophelia.  Ironically, Polonius gives such poor proof that Gertrude and Claudius insist on more covert spying!  Ha! Through this exchange of news, then, Polonius proves himself to be the fool of Hamlet and eventually, through his own foolhardiness, indirectly causes his own death behind the "arras."