Hi, I've been working on some Lebesgue measure and Lebesgue integral exercises for a few days and I have some doubts.
I need to say if the statements are true (I need to prove it) or false (I need to give a counterexample).
Let $f:E\subset\mathbb{R}^n\rightarrow\mathbb{R}^n$ such that $|f(x)-f(y)|\le|x-y|$ for $x,y\in\mathbb{R}^n$, then $f$ transforms null measure sets into null measure sets.
If two integrable functions agree in a dense set, the value of the integrals is the same.
If two functions agree in a dense set, one of then is measurable if and only if the other one is als