96 1. Measure theory is often referred to as convergence in mean, pointwise convergence almost everywhere is often referred to as almost sure convergence, and convergence in measure is often referred to as convergence in probability. Exercise 1.5.1 (Linearity of convergence). Let (X, B,μ) be a measure space, let fn,gn : X C be sequences of measurable functions, and let f, g : X C be measurable functions. (i) Show that fn converges to f along one of the above seven modes of convergence if and only if |fn f| converges to 0 along the same mode. (ii) If fn converges to f along one of the above seven modes of con- vergence, and gn converges to g along the same mode, show that fn + gn converges to f + g along the same mode, and that cfn converges to cf along the same mode for any c C. (iii) (Squeeze test) If fn converges to 0 along one of the above seven modes, and |gn| fn pointwise for each n, show that gn converges to 0 along the same mode. We have some easy implications between modes: Exercise 1.5.2 (Easy implications). Let (X, B,μ) be a measure space, and let fn : X C and f : X C be measurable functions. (i) If fn converges to f uniformly, then fn converges to f pointwise. (ii) If fn converges to f uniformly, then fn converges to f in L∞ norm. Conversely, if fn converges to f in L∞ norm, then fn converges to f uniformly outside of a null set (i.e. there exists a null set E such that the restriction fn X\E of fn to the complement of E converges to the restriction f X\E of f). (iii) If fn converges to f in L∞ norm, then fn converges to f almost uniformly. (iv) If fn converges to f almost uniformly, then fn converges to f point- wise almost everywhere. (v) If fn converges to f pointwise, then fn converges to f pointwise almost everywhere. (vi) If fn converges to f in L1 norm, then fn converges to f in measure. (vii) If fn converges to f almost uniformly, then fn converges to f in measure. The reader is encouraged to draw a diagram that summarises the logical implications between the seven modes of convergence that the above exercise describes.
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