PROBLEM SET 3

1. A function is called simple if it only takes on finite number of different values. If g is bounded and measurable, and ϵ > 0 is given, show there is a measurable simple function f so that sup_{x} |g(x) − f(x)| ≤ ϵ. Is this true if g is not bounded?

2. Suppose E is measurable set of real numbers and let f(t) = m(E ∩ (t − 1, t + 1)). Show that f is continuous.

3. Suppose E is a closed set in the upper half-plane whose vertical projection onto the real line is surjective (onto). For each real number x let y(x) be the closest point of E ∩ L_{x} to x (here L_{x} is the vertical line through x). Show that y is a measureable function, but need not be continuous.

4. Given a real number x ∈ [0, 1] let x_{n}(x) ∈ {0, 1} be its n-binary digit (if this is not unique, choose the epansion ending in all 0’s). Let f(x) = lim sup_{n→∞} \frac{1}{n} ∑_{k=1}^{n} x_{n}(x). Show that f is measureable. Where is f continuous? Can you guess what ∫_{0}^{1} f(x)dx is?