MTH644 Assignment 1 Solution Fall 2026

This assignment covers fundamental concepts in measure theory, specifically focusing on the properties of sigma-algebras, rings, and sigma-rings of sets. Students are presented with four distinct problems involving set definitions and specific conditions that subsets must satisfy. For each problem, students must perform mathematical verifications or proofs to determine if the given collections of sets meet the criteria for specific algebraic structures. The final output requires clear, step-by-step mathematical reasoning and derivations.

1. 1

   Question 1

   5 marks

   Given a set X and a collection of subsets H, verify whether H constitutes a sigma-algebra on X. You must provide a complete step-by-step solution with proper reasoning.

2. 2

   Question 2

   5 marks

   Given the set of integers X and two conditions for subsets A, determine if the resulting collection H is a ring of sets and whether it is a sigma-ring. Justify your answer with clear mathematical steps.

3. 3

   Question 3

   5 marks

   Given the set of real numbers X and specific conditions for subsets A involving intervals I and J, show that the collection H is a sigma-ring but not an algebra of sets on X. Provide full reasoning for your conclusion.

4. 4

   Question 4

   5 marks

   Given two sigma-algebras H1 and H2 on a set X, verify that both are indeed sigma-algebras. Then, determine if their union H1 ∪ H2 is also a sigma-algebra on X and justify your findings.

What to submit

One Microsoft Word (.doc or .docx) file using the built-in Equation Editor or Math Type for all mathematical expressions.

Topics covered

Sigma-algebras Ring of sets Sigma-rings Algebra of sets Set theory Countable sets Finite sets