I see the image contains several mathematical problems related to set theory, limits, and measure theory. Here's a brief overview of the questions visible:

1. Question 3:
   Given $A_1 = \{(x,y) \mid 0 < x < 2, 1 < y < 2\}$ and $A_2 = \{(x,y) \mid 1 < x < 3, 1 < y < 3\}$, determine $A_1 \cup A_2$ and $A_1 \cap A_2$.

2. Question 4:
   Given $\Omega = \{(x,y) \mid x^2 + y^2 \leq 2\}$ and $A = \{(x,y) \mid y \leq x\}$, determine $A^c$ (the complement of A).

3. Question 5:
   Given $\Omega = \{1, 2, 3, 4\}$ and a collection $A = \{ \emptyset, \{1\}, \{2\}, \{3\}, \{4\}, \{1,2\}, \{1,3\}, \{2,4\}, \Omega \}$, investigate if $A$ forms a σ-field or not.

4. Question 6:
   With $\Omega = \{1, 2, 3, 4\}$ and $A = \{ \emptyset, \{1,2\}, \{3,4\}, \Omega \}$, investigate whether $A$ forms a σ-field or not.

5. Question 7:
   If $\Omega = \{1, 2, 3, 4\}$ and $C = \{\{1\}, \{2\}\}$, determine $\sigma(C)$ (the σ-field generated by C).

Would you like detailed solutions to any specific questions?

Here are some follow-up questions for expanding your understanding:

1. What are the conditions for a collection of sets to form a σ-field?
2. How do you compute the union and intersection of regions defined by inequalities?
3. What is the complement of a set in terms of geometric regions?
4. How do you generate the σ-field from a collection of sets?
5. What are the concepts of lim inf and lim sup in the context of sequences of sets?

Tip: When determining if a collection is a σ-field, check if it satisfies closure under complementation and countable unions.