We now recall the fundamental convergence theorems relating limits and integration: the first three are for non-negative functions, the last three are for absolutely integrable functions. They are ultimately derived from their namesakes in Exercise 5 and an approximation argument by simple functions, and the proofs are again omitted.

Is there a typo in the numbering? I don’t see how the convergence theorems follow from Exercise 5. (It is probably Exercise 6.)

*[Corrected, thanks – T.]*

This is a reply to Mr. Jack Dippel. I’m having trouble understanding this counter-example. Can you help me understand why the space is sigma finite?

]]>assertion that when the measure is -finite, any element

of the -algebra is well-approximated by an element of

the algebra . For example, consider the case that

consists of the finite and co-finite subsets of

, and the measure is counting measure. Then the

extension theorem simply gives counting measure on all subsets of

; and if is neither

finite nor co-finite, then any set will have

.

*[You are right; I have deleted this part of the exercise. -T]*

The space of continuous functions (or, for that matter, Riemann integrable functions) is not closed under pointwise limits or infinite sums, whereas the space of measurable functions is. This is one of the main advantages of the Lebesgue integration theory in analysis, as it often allows one to interchange such limits and sums with integrals using such tools as the monotone convergence theorem or dominated convergence theorem. If one could only integrate functions that respected the topology (aka continuous functions), then it becomes significantly more difficult at a technical level to perform such interchanges.

]]>I am looking for Categorical properties of measurable space such as initial structure, final structure and discrete structure…Thanks in advance ]]>

I am a student newly admited into Msaters programme in mathematics but I have poor background in some knowledge of the basics more especially in measure theory and integration. I want you to please assist me with materials that will help me grasp the main ideas in measures. Thank you

Mustapha Abdussalam

Usmanu Danfodiyo University, Sokoto,

Nigeria.