1. Suppose that is a function such that is constant for almost all , and is constant for almost every . Prove that is constant ctp (with respect to u, where u is the Lebesgue measure).

Hint: Assume the contrary. Then it sets you and you have positive measure. Use Fubini to prove that each of these sets contains at least one vertical and one horizontal interval. Conclude.

Note: A function is constant ctp, if not constant in a set of measure zero.

For each y fixed

2.

4. Originally Posted by p33rz
Suppose that is a function such that is constant for almost all , and is constant for almost every . Prove that is constant ctp (with respect to u, where u is the Lebesgue measure).

Hint: Assume the contrary. Then it sets you and you have positive measure. Use Fubini to prove that each of these sets contains at least one vertical and one horizontal interval. Conclude.

Note: A function is constant ctp, if not constant in a set of measure zero.
What does f(.,y) denote?

5. For each y fixed

6. Originally Posted by p33rz
Suppose that is a function such that is constant for almost all , and is constant for almost every . Prove that is constant ctp (with respect to u, where u is the Lebesgue measure).

Hint: Assume the contrary. Then it sets you and you have positive measure. Use Fubini to prove that each of these sets contains at least one vertical and one horizontal interval. Conclude.

Note: A function is constant ctp, if not constant in a set of measure zero.

For each y fixed

7. Sorry. f must be such that:

is a function such that is constant for almost all , and is constant for almost every . Prove that is constant ctp (with respect to u, where u is the Lebesgue measure).

8. Originally Posted by p33rz
Sorry. f must be such that:

is a function such that is constant for almost all , and is constant for almost every . Prove that is constant ctp (with respect to u, where u is the Lebesgue measure).
, f(x,y) is constant for all y, the constant equals . And for a fixed y, f(x,y) is constant for every x.
Maybe I didn't understand what exactly you mean. Sorry for that. I am not from a English-speakiing country.

9. These look like homework questions p33rz. If so then you will battle to find help here unless you show us what you have tried already.

10. Originally Posted by river_rat
These look like homework questions p33rz. If so then you will battle to find help here unless you show us what you have tried already.
Yes.

Moreover, it appears to be a problem from an introductory graduate-level class in measure and integration, and outside help is such classes is often totally inappropriate. It would have been called outright cheating when I took the class years ago.

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