# I want to study about F sigma and G delta sets,Can someone suggest a book for learning them?

I am an undergraduate student of mathematics and I know nothing about F sigma and G delta sets.I want to learn and understand these concepts along with their applications.Can anyone suggest me a good book that would be helpful for me?

The basic definition is that in a topological space $$\Omega$$, an $$F_{\sigma}$$ set is a countable union of closed sets and a $$G_{\delta}$$ set is the complement of an $$F_{\sigma}$$ set, i.e. a countable intersection of closed sets (using De Morgan).

I have only encountered $$F_{\sigma}$$ and $$G_\delta$$ sets in Measure Theory, where they play a natural role, since they are (almost by definition) contained in the Borel $$\sigma$$-algebra of $$\Omega$$ (the $$\sigma$$-algebra generated by open sets).

The most interesting application I have seen involving $$F_{\sigma}$$ and $$G_\delta$$ sets is to understand the structure of a Lebesgue measurable set on $$\mathbb{R}^d.$$ Namely, there is a proposition that states that any Lebesgue measurable set on $$\mathbb{R}^d$$, is "basically" an $$F_{\sigma}$$ and a $$G_\delta$$ set. The "basically" here means "up to a null set" (a set of measure $$0$$). This fact is also related to Littlewood's three principles of real analysis.

A good book to read about the things I talked about is Terry Tao's "Introduction to Measure Theory".

Bruckner, Bruckner and Thompson's Real Analysis is a rather good book for these kinds of sets: they introduce them in the first chapter (starting at page 5!) and provide a good breakdown of how they develop $$F_{\sigma \delta}, F_{\sigma \delta \sigma}, \ldots$$ and the inclusions, and they provide structured exercises to further improve your understanding.

The book might prove quite useful throughout your undergraduate course as well as it covers metric spaces, measurable functions, Baire category and Banach and Hilbert spaces.

It's worth noting, in light of another poster's answer, that these sets are used to illuminate the depths of Real Analysis, not just measure theory. They form the foundations of the Borel sets, which cover just about any set you encounter in an undergraduate course (because they contain all open and closed sets, all countable unions of such, and all complements of such). When you start asking if the Borel sets are everything (they're not!) then you start to uncover the pathological cases that allow you to properly understand some of the convergence theorems you learn about.