Reference no: EM132375540

Measure Theory and Fourier Analysis (Advanced) Assignment -

1. Let n ≥ 3. Construct a subset of C₀ := [0, 1] as follows. Remove the open interval of length 1/n from the middle of C₀. This leaves a set C₁ consisting of two closed intervals of equal length. Then remove open intervals of length 1/n² from the middle of each interval in C₁, leaving the set C₂ consisting of four intervals of equal length. Continue that procedure, removing open intervals of length 1/n^k in the k-th iteration. Finally define

C := _k=1∩^∞C_k.

As an intersection of closed sets C is closed.

(a) Show that C has Lebesgue measure n-3/n-2.

(b) Show that C does not contain any open interval.

(c) Use the set C ⊆ R to show that in any dimension N ≥ 1 there exist closed bounded sets in RN with empty interior and positive measure. For N ≥ 2, how can you make such a set connected, that is, between every pair of points there is a continuous curve in the set that connects the two points.

2. The Lebesgue outer measure m*(A) of a subset A ⊆ R^N is defined by

m*(A) := inf{_k=0Σ^∞vol(R_k): R_k are open rectangles and A ⊆ _k=0U^∞R_k}

with vol(R_k) being the usual volume of a rectangle. For N ≥ 2, use this definition to show that m*_N(R^N-1 x {0}) = 0.

3. Let (X, A, μ) be a measure space and let f_k : X → C be measurable. Assume that

_k=1Σ^∞∫_X|f_k|dμ < ∞

converges. Prove that

_k=1Σ^∞f_k(x)

converges absolutely almost everywhere (convergence means a limit in C, not ±∞).