Measure spaces, measurable functions and positive measures, integration, convergence*. Examples. 1) fn(x) = n−1χ(0,n)(x), 2) fn(x) = χ(n,n+1)(x), 3) fn(x) = nχ[0,1/n](x), 4) L1 convergent but oscillating. Definition. pointwise convergence, uniform convergence, and convergence in Lp . Definition. We say that { fn} of measurable complex-valued functions on (X ,M ,μ) is Cauchy in measure if for ev...

If φ is an analytic selfmap of the disk (not an elliptic automorphism) the DenjoyWolff Theorem predicts the existence of a point p with |p| ≤ 1 such that the iterates φn converge to p uniformly on compact subsets of the disk. Since these iterates are bounded analytic functions, there is a subset of the unit circle of full linear measure where they are all well-defined. We address the question o...