I am reading Rudin's RCA and have a question regarding the proof of Jensen's inequality.
Let be a positive measure on a -algbebra in a set , so that . If is a real function in , if for all , and if is convex on then
After a few steps Rudin obtains the following inequality:
for every . Here, and is a real number. If then the inequality can be obtained by integrating both sides. However, I am not sure how to proceed in the case . Rudin states that in this case, the integral is defined in the extended sense
I am unable to show the existence of as defined above.
Any help would be greatly appreciated. Thank you.