I still have no background in measure theory or Lebesgue integration. So is there any elementary-level resource --- book, set of notes, tutorial paper, website, or online video lecture series --- which I can use for getting some knowledge of these fields?
I would like to suggest three books which helped me-
1) Royden's Real Analysis,here in this it gives motivation towards the topic as well as illustrative text,nice examples,excercises.
2)Measure Theory and Integration by G. de Barra.
3)Paul Halmos,Measure theory.
Hope this helps!
This is an introductory video course on measure and integration by Prof.Inder Rana of IIT Bombay. There are total 40 lectures with each one almost of an hour. Also he uses his own book on measure and integration along with Paul Halmos's and Royden's as mentioned in @BAYMAX's answer.
The syllabus of the course as well as the references are mentioned in the link.