I find that when learning more abstract concepts, it helps to have a 'simple' example tied to every theorem in order to fully appreciate the theorem or property.
However, the course notes I am currently using for my studies of Measure Theory (this is a first course for me) do not have many examples. It is highly rigorous and detailed, but lacks examples.
Are you aware of any introductory books on Measure Theory that has a good number of examples tied to each theorem? In other words I guess, do you know of very 'gentle' books on this topic?
I usually refer the books 'Measure theory and Integration' by G De Barra and 'An Introduction to Measure and Integration' by Inder K. Rana which are having a number of examples and exercise problems.