Lebesgue Integral on $\mathbb{R}$ vs Lebesgue Integral on $\mathbb{R}^n$

Question

Now I am finshing the chapters of Royden about Lebesgue integration (Lebesgue Integral on $\mathbb{R}$) . My personal goal is to learn Integration on general measure spaces. I would like to ask:

1)It is necessary to continue my study from $\mathbb{R}$ to $\mathbb{R}^n$ and after on general spaces or immediately to general/abstract Spaces?

2)Is there any advantage or disadvantage folowing this patern of study?

I am asking this because I have found a lot of Books with the one or the another approach (Stein, Carothers, Folland, Zygmund, Frank Jones, ...)

Any suggestions are welcome,

Thanks!

Answer 1

I think measures on abstract spaces are useful in almost every branch of Mathemetics. If you have studied Lebesgue integral on $\mathbb R$ you should head straight towards abstract measure spaces. When you do that you will learn what are called product measures which gives you one way of constructing Lebesgue measure on $\mathbb R^{n}$. So you can study Lebesgue measure on $\mathbb R^{n}$ at that stage.