MATH 395 PROBLEMS 11

IGOR KRIZ

Regular problems:

1. Calculate

∫_{Q} f(x + y)dxdy

where Q = [0, 2] × [0, 2] and f(t) denotes the greatest integer ≤ t. [Find some way to visualize the function; the fundamental theorem of calculus won’t work.]

2. Prove that if S is a countable set, then the set of all finite subsets of S is countable.

3. Using theorems from class (your notes), prove that if f_{n} : [a, b] → ℝ are

continuous functions and f_{n} ↗ f, then f is continuous if and only if f_{n} \overrightarrow{→} f.

4. (a) Calculate

∫_{0}^{1} ln(x)dx,

and justify using Lebesgue integral.

(b) Prove, more generally, that if f : [a, b] → [0, ∞) (or f : [a, b] → (−∞, 0] and f is continuous on [a, b), then

∫_{a}^{b} f = lim_{x→a} ∫_{a}^{x} f

where ∫ denotes Lebesgue integral. [Hint: express f as an increasing (resp. decreasing) limit of continuous functions.] Challenge problems:

5. The Cantor set C is the set of all x ∈ [0, 1] which are not of the form

\frac{k + α}{3^{n}}

for any k ∈ ℤ, α ∈ (1/3, 2/3). Define a function f : ℝ → ℝ by

f(x) = \{ \begin{matrix}1 if x ∈ C \\ 0 else. \end{matrix}

Is f Lebesgue-integrable? If yes, what is the integral equal to?

6. Prove that the set of all subsets of ℕ is not countable. [Find a method analogous to the one used for the real numbers.]