We have Riemann integration in our next semester, for which I need to cover basic ideas of measure or length of a set in $\mathbb R$. Can anyone tell me how much do I need to know about measure theory or what parts are sufficient to cover to understand the Riemann integration?
Also can anyone suggest me some reference materials to cover up those essential topics of measure theory?
(Note that I must cover only those things which are essential and in a very short time - say one week. I am not going to study the entire measure theory and secondly I know almost nothing of measure theory).
Depends on your course's syllabus. For a quick reference you can check this: https://terrytao.files.wordpress.com/2011/01/measure-book1.pdf by Terence Tao. Personally I studied by Folland's Real Analysis during my Real Analysis (Measure Theory) course, it's a bit wordy, though clear and good. Hard way: W. Rudin - Real and Complex Analysis.