Book recommendation on Integration (Riemann, Lebesgue, Riemann-Stieltjes, Ito) I am interested in an advanced calculus book that focuses primarily on integration. Ideally it covers the basics of Riemann integral, rules of differentiation and integration under the Riemann, then leaps to the Lebesgue measure and integration, covers the Riemann-Stieltjes integral and Ito integral.
Please share your recommendations.
Thanks!
 A: For an introduction to Riemann integration, I would recommend either Abbott or Rudin's Real Analysis. This will cover a bit more than just integration, but it will help to know properties of $\mathbb{R}$, continuity and metric spaces for studying Riemann integration. Of these, Abbott spends a bit more time on exposition and so I would recommend that first.
Going for Lebesgue integration, there are a few resources that I can recommend:
1) Tao's Measure Theory
2) Royden's Real Analysis
These will cover Lebesgue integration amply, but I think that they have limited coverage of the RS-integral.
The Riemann-Steljtes integral and Lebesgue measure is well covered in a less-well-known textbook by Torchinsky known as Real Variables. This book introduces measure abstractly first before deciding to focus on the Lebesgue measure. This means that you will see important theorems first in an abstract measure space before restricting to $\mathbb{R}$. Whether or not this suits you is dependent on whether you are interested in working in other measure spaces. RS-integrals are, in my mind, tied to BV functions and monotonic functions, so there will be some effort needed for "setting the scene".
The Ito integral is covered more in texts about stochastic calculus. 
A: You might find A Garden of Integrals by Frank E. Burk interesting since it is one of the only texts I know to deliberately cover a plethora of integrals (or a garden even). This is not quite a recommendation as I have not read it.
