Real variable function is a difficult course. My feeling is that it is very important to see the definition clearly and think clearly about the routine. Why is definition important? Compared with Riemann integral, the construction of Lebesgue integral is more complicated. When studying Riemann integral, in fact, many definitions involved have been studied directly or indirectly in high school, so it seems that you can master the definitions well without paying too much attention. However, Lebesgue integral is different. Although it is a generalization of Riemann integral, it involves a wide range, and many concepts (definitions) have not been discussed before, such as the introduction of measure. If you want to learn this course well, it is impossible without a clear definition. I suggest that every time you come across a new definition, ask yourself, ask the book and ask the teacher why you need it. It is necessary to have a definition. Of course, understanding the definition is the micro-level in learning, and the macro-level needs to understand the routine of the overall construction of Lebesgue integral: how to start from set theory, introduce measure, define convergence, define simple function integral and then general function integral. In short, in the process of learning, you need to constantly ask why at the micro level and how to do it at the macro level. If you can grasp these two points, I think you can learn this course well.

In addition, reading books on measurement theory after class is good for learning real variable functions.