I am new to the Stackexchange community so do let me know if I can improve my question in any way.
Right, I have just started reading Michael Lacey's proof of Carleson's theorem (http://people.math.gatech.edu/~lacey/research/esi.pdf), and have already hit a problem.
In Proposition 1.4 (Page 4), I don't understand a few things:
1) Why has the author proceeded in this method for establishing that the desired set is closed?
My instincts would have said, take a sequence in the set of functions for which the results hold, say they converge in the norm to some function and finally prove that the said function is member of the desired set.
I think I can, however, see how the two might be equivalent: if one proves the desired set is closed the criterion that Lacey is trying to establish follows. Vice-versa if one proves what Lacey sets out to prove in the first line of the proof then the set must be closed.
2) This is just to confirm my own instincts, but I believe the result follows from an application of Chebyshev's Inequality.
I would be grateful for any help/pointers and as above please don't hesitate to let me know if I haven't complied with any other rules of the forum.