I am a graduate student of Mathematics.I am self-studying measure theory.I have already completed measure,integration,convergence theorem etc.I am yet to study product measures but I am not finding any book suitable for that topic.I have referred to the following books:
$1.$ S.Kesavan :Measure and integration
$2.$ Patrick Billingsley:Probability and measure
But the problem with these books is that they are too abstract to study.For example,the second one defines product measure by introducing $\pi$-system and $\lambda$-system.I am finding a book that introduces product measure in a beginner-friendly way.Can someone suggest suitable reference for this?
I think using the $\pi$-$\lambda$ theorem is the simplest and most direct way to introduce the product measure on a product of two finite measure spaces $(X, \mathcal{F}, \mu)$, $(Y, \mathcal{G}, \nu)$. Here is a sketch:
For $C \in \mathcal{F} \otimes \mathcal{G}$, let $$(\mu \times \nu)(C) := \int_{X}\int_{Y}1_C(x, y)\nu(dy)\mu(dx)$$ Use the $\pi$-$\lambda$ theorem to show that this integral is well defined , i.e. that the integrand $x \mapsto \int_{Y}1_C(x, y)\nu(dy)$ is measurable. The uniqueness of the product measure is an easy application of the $\pi$-$\lambda$ theorem. So you immediately get $\int_{X}\int_{Y}1_C(x, y)\nu(dy)\mu(dx) = \int_{Y}\int_{X}1_C(x, y)\mu(dy)\nu(dx)$, known as Tonelli's theorem.
From the finite case, you easily get a product measure in the case where $\mu$ and $\nu$ are $\sigma$-finite, and uniqueness of the product measure follows from the $\pi$-$\lambda$ theorem.
This approach is certainly less abstract than Caratheodory's theorem. Also the $\pi$-$\lambda$ theorem has many other uses, so it's worth knowing. The downside is that it seems that you need Caratheodory's theorem to get a product measure on an infinite product of probability spaces.
Klenke's probability book takes this approach to defining finite product measures in chapter 14. Though the chapter (and book in general) is somewhat abstract. I found https://services.math.duke.edu/~rtd/PTE/PTE5_011119.pdf to be more readable, thought it doesn't spell out the above approach.