Self-Study Analysis: Metric Spaces, Measure Theory & More

Direct answer: After finishing analysis on real numbers and n-dimensional real space, the recommended path for self-study is Carothers’ Real Analysis for metric spaces, function spaces, and measure theory on the real line, followed by Jones’ and Bartle’s texts for deeper measure theory, and Kreyszig’s Introductory Functional Analysis with Applications to begin functional analysis.

Key Takeaways

• N. L. Carothers’ Real Analysis is the recommended starting text, covering metric spaces, function spaces, and measure theory on the real line $\mathbb{R}$.
• Frank Jones’ Lebesgue Integration on Euclidean Space extends measure theory to $\mathbb{R}^n$ and includes a formal proof connecting the determinant to volume.
• Robert G. Bartle’s The Elements of Integration and Lebesgue Measure presents the same material as Jones but from a more abstract, general measure-space perspective.
• Erwin Kreyszig’s Introductory Functional Analysis with Applications requires no prior measure theory and can be started alongside the metric and function spaces sections of Carothers.
• The Henstock-Kurzweil (gauge) integral, covered in Bartle’s A Modern Theory of Integration, is optional but stronger than the Lebesgue integral in some respects on $\mathbb{R}$.

What background is needed before starting this study path?

This study path assumes familiarity with analysis on the real numbers $\mathbb{R}$ and on n-dimensional real space $\mathbb{R}^n$. A separate insight lists the prerequisite topics and book suggestions for that foundation: Self-Study Analysis: Introduction to Analysis.

Comfort with linear algebra is also required, since functional analysis later generalizes linear algebra to infinite-dimensional spaces. A dedicated insight covers linear algebra self-study: Self-Study Algebra: Linear Algebra.

Why move beyond analysis on R and R^n?

Analysis on $\mathbb{R}$ and $\mathbb{R}^n$ is only an introduction to a much larger subject. A student who has learned that material well should find the next stage manageable, since most of it consists of careful, powerful generalizations of the classical results already studied.

The concepts of metric spaces, normed spaces, and Hilbert spaces on one side, and measure theory on the other, form the backbone of nearly all further analysis. These two threads are the core focus of the books recommended below.

Which book introduces metric spaces, function spaces, and measure theory?

Carothers, Real Analysis

N. L. Carothers’ Real Analysis is recommended as the starting point for this stage of study. The book motivates new concepts carefully, provides substantial intuition at every step, and includes excellent exercises, with some problems tagged “must do” and others left optional. The book can be found here: Carothers, Real Analysis (Amazon).

Carothers covers metric spaces, function spaces, and measure theory on $\mathbb{R}$, with thorough and deep treatment of each. Its one limitation is that it does not develop measure theory on more general spaces beyond $\mathbb{R}$.

What does the metric spaces section of Carothers cover?

The metric spaces section of Carothers’ Real Analysis covers open and closed sets, continuity, completeness, compactness, connectedness, and category theorems.

For a more thorough treatment of metric spaces specifically, M. O’Searcoid’s Metric Spaces is a useful supplement: O’Searcoid, Metric Spaces (Amazon).

What does the function spaces section of Carothers cover?

The function spaces section of Carothers’ Real Analysis covers uniform convergence, Fourier series, the Stone-Weierstrass theorem, the Ascoli-Arzelà theorem, bounded variation, and Stieltjes integration.

What does the measure theory section of Carothers cover?

The measure theory section of Carothers’ Real Analysis covers Lebesgue measure, measurable functions, the Lebesgue integral, and differentiation theory, all developed on the real line $\mathbb{R}$.

Which book deepens measure theory on R^n?

Jones, Lebesgue Integration on Euclidean Space

Frank Jones’ Lebesgue Integration on Euclidean Space gives a deep but intuitive development of measure theory on $\mathbb{R}^n$ and sometimes addresses general measure spaces. Where many texts rush the construction of Lebesgue measure, Jones proceeds carefully and cleanly, and the book discusses several applications of measure theory within analysis. It is available here: Jones, Lebesgue Integration on Euclidean Space (Amazon).

This book covers Lebesgue measure on $\mathbb{R}^n$, the invariance of Lebesgue measure with a formal proof of why the determinant measures volume, Borel sets and measurable functions, Lebesgue integration over $\mathbb{R}^n$, the Fubini-Tonelli theorem, the Gamma function, $L^p$ spaces, convolutions, Fourier theory on $\mathbb{R}^n$, and differentiation theory on both $\mathbb{R}^n$ and $\mathbb{R}$.

Which book covers measure theory on general spaces?

Bartle, The Elements of Integration and Lebesgue Measure

Robert G. Bartle’s The Elements of Integration and Lebesgue Measure covers much of the same territory as Jones’ Lebesgue Integration on Euclidean Space but from a more abstract perspective, and it is a good idea to work through both books concurrently. The book is available here: Bartle, The Elements of Integration and Lebesgue Measure (Amazon).

Bartle’s book addresses measurable functions, measures, the integral, integrable functions, $L^p$ spaces, different kinds of convergence, decomposition of measures, generation of measures, product measures, outer measure, measurable sets, approximation of measurable sets, additivity, and non-Borel and non-measurable sets.

After completing Bartle’s text, a student will have sufficient familiarity with measure theory to read most advanced analysis texts.

Is the Lebesgue integral the most general integral?

The Lebesgue integral is not the most general integral available. On $\mathbb{R}$ there is the Henstock-Kurzweil (gauge) integral, which is stronger in some respects. Studying it is optional, but readers who wish to explore it can consult Bartle’s A Modern Theory of Integration: Bartle, A Modern Theory of Integration (Amazon).

Which book introduces functional analysis?

Kreyszig, Introductory Functional Analysis with Applications

Erwin Kreyszig’s Introductory Functional Analysis with Applications is a standard, elementary introduction to functional analysis, available here: Kreyszig, Introductory Functional Analysis with Applications (Amazon).

This book does not require measure theory, so it can be read after the metric and function spaces parts of Carothers’ Real Analysis, and it is also natural to study it alongside Carothers. Readers who have already studied measure theory will still find Kreyszig valuable, since it is a useful exercise to try to extend its theorems to more general contexts.

What does Kreyszig cover?

Kreyszig’s Introductory Functional Analysis with Applications covers metric spaces, normed spaces and Banach spaces, inner product spaces and Hilbert spaces, fundamental theorems for normed and Banach spaces, the Banach fixed-point theorem with applications to linear, differential, and integral equations, approximation theory, spectral theory of linear operators in normed spaces, compact linear operators on normed spaces and their spectrum, spectral theory of bounded self-adjoint linear operators, and unbounded linear operators in Hilbert space, including applications to quantum mechanics.

Many of these topics overlap with Carothers’ Real Analysis but are treated from a different perspective. The book closes with a section on quantum mechanics that helps motivate the subject of functional analysis, which generalizes linear algebra to infinite-dimensional settings.

What comes after this stage?

The next recommended steps after this stage of study are complex analysis and topology, covered in a separate insight.

Glossary

• Metric space – a set equipped with a distance function satisfying specific axioms, used to define concepts like continuity and completeness.
• Normed space – a vector space equipped with a norm, a function that assigns a length to each vector.
• Hilbert space – a complete normed space whose norm comes from an inner product.
• Measure theory – the branch of analysis that generalizes the notion of length, area, and volume to abstract sets.
• Lebesgue integral – an integral defined using measure theory that generalizes the Riemann integral.
• Borel set – a set that can be formed from open sets through countable unions, intersections, and complements.
• L^p space – a space of functions whose p-th power of the absolute value is integrable.

Frequently Asked Questions

What should I study before starting metric spaces and measure theory?

Before starting this stage, be comfortable with analysis on the real numbers $\mathbb{R}$ and on n-dimensional real space $\mathbb{R}^n$, as well as linear algebra. These foundations are covered in separate insights on analysis and linear algebra self-study.

Do I need measure theory before starting functional analysis?

No. Kreyszig’s Introductory Functional Analysis with Applications does not require measure theory, so it can be started after or alongside the metric and function spaces sections of Carothers’ Real Analysis.

Should I read Jones or Bartle first for measure theory?

The recommendation is to work through Jones’ Lebesgue Integration on Euclidean Space and Bartle’s The Elements of Integration and Lebesgue Measure concurrently, since Jones focuses on $\mathbb{R}^n$ while Bartle treats the same material more abstractly.

Is the Henstock-Kurzweil integral necessary to learn?

No, studying the Henstock-Kurzweil (gauge) integral is optional. It is stronger than the Lebesgue integral in some respects on $\mathbb{R}$, and interested readers can pursue it through Bartle’s A Modern Theory of Integration.

What topics come after metric spaces, measure theory, and functional analysis?

The next recommended subjects are complex analysis and topology, which are addressed in a separate insight rather than this one.