Any help with any of these would be very much appreciated, even if it's just a link to some useful material, etc. Thanks in advance!
- Let be a -finite measure space. Suppose that is a -measurable function such that for every integrable function . Prove that -almost everywhere on .
- Let be a -finite measure space. Let be a -measurable function with and let . Prove that for every with , .
- Let be a -finite measure space. Let be integrable. Prove that there exists a bounded -measurable function such that and .
- If , prove that -almost everywhere. Moreover, prove that if , then there exists an with and for all .
- Prove that is a normed linear space.