PROBLEM SET 3

1. If f is measurable, show that the set of local maxima of f is a measureable set (x is a local maximum of f if there is an interval I centered at x so that f(x) = max_{y∈I} f(y)).

2. Suppose f is continuous on the reals and let f^{(}2) = f ∘ f and f^{(n)} = f ∘ f^{(n−1)} = f ∘ f ∘ … ∘ f n times. Let F (x) = 0 if {x_{n}} = {f^{n}(x)}_{n=1}^{∞} is bounded and F (x) = 1 if the sequence is unbounded. Show F is measurable.

3. 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?