Real Analysis, Math 821.

Instructor: Dmitry Ryabogin

Assignment II.

1. Problem 1. Let S be a collection of sets. Prove that there exists one and only one minimal ring R(S), such that S ⊂ R(S). (This, in particular, means that for any ring R containing S, R(S) ⊂ R).

Hint. Define X := ∪_{A∈S}A, and consider P (X) (the set of all subsets of X). Let Σ be a collection of all rings containing S and which are contained in P (X). The intersection of all of these rings (elements of Σ) will be your minimal ring.

2. Problem 2^{∗}. Let S be a subring. Prove that the minimal ring R(S) is a collection of sets of the type A = ∪_{k=1}^{n}A_{k}, A_{k} ∈ S.

Hint. Prove by induction (and then use) the following statement.

Lemma. Let A_{i}, A, i = 1, ..., n belong to a subring S, A_{i} ⊂ A, and let A_{i} ∩ A_{j} = ∅ for i ≠ j. Then there are sets A_{n+1}, ...A_{s} belonging to S, such that A = ∪_{k=1}^{s}A_{k}, s ≥ n.

3. Problem 3. Let (E_{n})_{n=1}^{∞} be a sequence of sets. The set \overline{lim_{n→∞}}E_{n} := ∩_{n=1}^{∞}( ∪_{k=n}^{∞} E_{k})

is called the upper limit of the sequence. The set \underline{lim}_{n→∞} E_{n} := ∪_{n=1}^{∞}( ∩_{k=n}^{∞} E_{k}) is called

the lower limit of the sequence. Prove that \underline{lim}_{n→∞} E_{n} ⊆ \overline{lim_{n→∞}}E_{n}. If they are equal, then

their value (the set) is called a limit of sequence of sets.

4. Problem 4. Give an example of sets for which \underline{lim}_{n→∞} E_{n} ≠ \overline{lim_{n→∞}}E_{n}.

5. Problem 5. Let X be a set, and let (E_{n})_{n=1}^{∞} be a sequence of sets satisfying E_{n} ⊂ X

for n = 1, ..., ∞. Prove

X \ \overline{lim_{n→∞}}E_{n} = \underline{lim}_{n→∞} (X \ E_{n}).

6. Problem 6. Let (E_{n})_{n=1}^{∞} be a sequence of sets and let (1_{E_{n}})_{n=1}^{∞}(x) be a sequence of their characteristic functions. Prove that

1_{\{\overline{lim_{n→∞}}E_{n}\}}(x) = \overline{lim_{n→∞}}1_{E_{n}}(x), 1_{\{ \underline{lim}_{n→∞} E_{n}\}}(x) = \underline{lim}_{n→∞} 1_{E_{n}}(x).