Solutions to Bogachev's Measure Theory, Volume 1

Sun, 23 Aug 2026 20:10:00 +1000

Solutions to the exercises of V.I. Bogachev’s Measure Theory, Volume 1 (Springer, 2007) — exercises 1.12.47–160, 2.12.25–117, 3.10.29–125, 4.7.42–154, and 5.8.37–142. Exercises marked \(\circ\) in the book are the basic ones. The book itself is filed at Measure Theory (Bogachev).

Constructions and Extensions of Measures

The Lebesgue Integral

Operations on Measures and Functions

The Spaces \(L^p\) and Spaces of Measures

Connections between the Integral and Derivative

Solutions to Kolmogorov and Fomin's Introductory Real Analysis

Sun, 20 Jul 2025 02:42:32 +1100

Set Theory

Sets and Functions

Problem

Prove that if \(A\cup B = A\) and \(A \cap B = A\), then \(A = B\).

Solution

To show \(A=B\), show \(A\subseteq B\) and \(B\subseteq A\). Suppose \(x\in B\), then we know by definition that \(x\in (A\cup B)\) if \(x\in B\) or \(x\in A\). Which then implies that \(x\in A\) from rule 1. Thus \(B\subseteq A\). Now suppose \(x\in A\) which implies \(x\in (A\cap B)\) (rule 2). The definition of this means that \(x\in A\) and \(x\in B\). \(\therefore x\in A \implies x\in B\), i.e. \(A\subseteq B\), so \(A=B\).

Problem

Show that in general \((A-B)\cup B \neq A\).

Solution

This only holds for \(B\subseteq A\). We proceed by counterexample. Let \(A={1,2}, B={3,4}\). Then \((A-B) = {1,2}\) and \((A-B)\cup B = {1,2,3,4} \neq {1,2}\).

Problem

Let \(A = {2,4,…,2n,…}\) and \(B = {3,6,…,3n,…}\). Find \(A\cap B\) and \(A-B\)

Solution

\[ A\cap B = {6n \mid n\in \mathbb{N}} \] \[ A - B = {2n \mid n\in \mathbb{N}, 2n \not\in {6m \mid m\in \mathbb{N}}} \]

Problem

Prove that:

()

\((A-B)\cap C = (A\cap C) - (B\cap C)\)

Solution:

Let \(x\in (A-B)\cap C\). Then:

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Solutions to Bogachev's Measure Theory, Volume 1

Constructions and Extensions of Measures#

The Lebesgue Integral#

Operations on Measures and Functions#

The Spaces \(L^p\) and Spaces of Measures#

Connections between the Integral and Derivative#

Solutions to Kolmogorov and Fomin's Introductory Real Analysis

Set Theory#

Sets and Functions#

Constructions and Extensions of Measures

The Lebesgue Integral

Operations on Measures and Functions

The Spaces \(L^p\) and Spaces of Measures

Connections between the Integral and Derivative

Set Theory

Sets and Functions