Real Analysis, Math 821.

Instructor: Dmitry Ryabogin

Assignment VI.

1. Problem 1.

a) Let f : R^{n} → R be a continuous function, and let g_{1}(x), g_{2}(x), ..., g_{n}(x) be measurable functions (defined on some space X). Prove that h(x) := f(g_{1}(x), g_{2}(x), ..., g_{n}(x)) is measurable.

Hint. The set {(t_{1}, ..., t_{n}) ∈ R^{n} : f(t_{1}, ..., t_{n}) > a} can be written as a union of open sets of the type (a_{k1}, b_{k1}) × ... × (a_{kn}, b_{kn}), k = 1, 2, ... Then

{x ∈ R^{n} : h(x) > a} = ∪_{k=1}^{∞} ∩_{i=1}^{n} {x ∈ R : a_{ki} < x < b_{ki}}.

b) A function f : X → C, f(x) = u(x) + iv(x), is called measurable if u(x), v(x) are measurable. Prove that |f(x)|, and arg f(x) are measurable.

c) A vector-function f : X → R^{n} is called measurable if the coordinate functions (f_{1}(x), f_{2}(x), ..., f_{n}(x)) of the vector f(x) are measurable with respect to some fixed basis in R^{n}. Prove that this definition does not depend on the choice of the basis.

d) Let f : X → R. Describe a ∈ R such that measurability of f(x)^{a} implies the measurability of f(x).

2. Problem 2.

a) Let f(x) be a differentiable function on [0, 1]. Prove that f′(x) is measurable.

b) Let (f_{n}(x))_{n=1}^{∞} be a sequence of measurable functions. Prove that sup_{n} f_{n}(x), inf_{n} f_{n}(x)

are measurable.

c) Let (f_{n}(x))_{n=1}^{∞} be a sequence of measurable functions. Prove that \overline{lim}_{n} f_{n}(x), \underline{lim}_{n} f_{n}(x)

are measurable.

3. Problem 3.

a)* Let f(x) be continuous on [0, 1], and let N_{f}(c) be a function, (called the indicator of Banach), defined as a number (possibly infinite) of solutions of the equation f(x) = c. Prove that N_{f} is measurable.

Hint. For each n ∈ N divide [0, 1] into segments [(j − 1)2^{−n}, j2^{−n}], j = 1, 2, ..., 2^{n}, and define N_{n}(c) as a number of segments [(j − 1)2^{−n}, j2^{−n}] containing at least one solution f(x) = c. Show that functions N_{n}(c) are measurable, and prove that N_{f}(c) =

lim_{n→∞} N_{n}(c).

b) Let f(x) be a one-to-one mapping of [0, 1] onto [0, 1] × [0, 1] defined as follows: let x = x_{1}, x_{2}, x_{3}, ...x_{n}, ... ∈ [0, 1] be binary-irrational (this means that among x_{i} we have infinitely many 0 and 1). Then, f(x) := (y_{1}, y_{2}), where y_{1} = (x_{1}, x_{3}, x_{5}, ...), and y_{2} = (x_{2}, x_{4}, x_{6}, ...). For binary-rational numbers choose one way of writing them in terms of 0 and 1, and define f the same way.

Prove that f is Lebesgue measurable and preserves the measure ( if B ⊂ [0, 1] × [0, 1] is Lebesgue measurable, then f^{−1}(B) is Lebesgue measurable on [0, 1], and M(B) ≡ m(f^{−1}(B)), where m is Lebesgue measure on [0, 1], and M is Lebesgue measure on [0, 1] × [0, 1]).

c) Let x = 0, n_{1}n_{2}n_{3}, ..., y = 0, m_{1}m_{2}m_{3}, ..., decimal decompostions of x, y ∈ [0, 1]. Define f(x, y) = k provided n_{k} = m_{k} and n_{i} ≠ m_{i} for i < k; f(x, y) = ∞ provided n_{k} ≠ m_{k} for all k. Prove that f is Lebesgue measurable and finite almost everywhere.

4. Problem 4.

a) Let f_{n} → f and f_{n} → g almost everywhere as n → ∞. Prove that f is equivalent to g.

b) Let (r_{k})_{k=1}^{∞} be a sequence of rationals from [0, 1], and let r_{k} = p_{k}/q_{k} be irreducible fraction. Define f_{k}(x) := exp{−(p_{k} − xq_{k})^{2}}. Prove that f_{k} → 0 as k → ∞ in measure, but lim_{k→∞} does not exist at any point x ∈ [0, 1].

Hint. To show that the pointwise limit does not exist proceed as follows. Take any x ∈ [0, 1] and consider a sequence (r_{k_{n}})_{n=1}^{∞} such that lim_{n→∞} r_{k_{n}} exists and not equal to x. Then show that f_{k_{n}}(x) → 0. On the other hand, there exists a sequence (r_{k′}_{n})_{n=1}^{∞} such that |p_{k′}_{n}/q_{k′}_{n} − x| ≤ 1/q_{k′}_{n}. Prove that f_{k′}_{n}(x) ≥ e^{−1}.

c) Write out a subsequence (k_{l})_{l=1}^{∞} such that lim_{l→∞} f_{k_{l}}(x) = 0 almost everywhere.

5. Problem 5*. We saw in class that

D(x) = lim_{n→∞} lim_{m→∞}(cos(2πn!x))^{m}

where D(x) is the Dirichlet function (taking 0 at irrationals, and 1 at rationals).

a) Prove that you may not write D(x) = lim_{n→∞} φ_{n}(x) where φ_{n}(x) continuous functions.

Hint. Let F_{n} := \{x ∈ R : φ_{n}(x) ≤ 1/2\}. Then \{x ∈ R : D(x) < 1/2\} = \underline{lim}_{n→∞} F_{n}.

But this contradicts the fact that irrationals may not be written as a countable union of closed sets.

b) Conclude that it is impossible to define a metric d on the space X of all functions f : R → R such that the convergence in d would be equivalent to the pointwise convergence.

Hint. Assume the contrary, we have a metric d, such that f_{n}(x) → f(x) pointwise is the same as d(f_{n}, f) → 0 as n → ∞. Let (g_{m}(x))_{m=1}^{∞}, (f_{m,n}(x))_{m,n=1}^{∞} be such that d(f_{m,n}, g_{m}) → 0 as n → ∞, and d(g, g_{m}) → 0 as m → ∞. But then ∀m ∈ N there exists n(m) ∈ N such that d(f_{m,n(m)}, g) → 0 (use the diagonal process of Cantor). This contradicts a) (why?).