I need help for a very basic problem. I recognize it should be immediate and easy however, I'm somehow stuck and I am wondering if I really understood the exercise or the definitions, so please, don't be rude with me.
Let $ \mu = \lambda + i\lambda$ where $\lambda$ denotes the Lebesgue measure on $ [0,1] $.
We want to find a set $A\subset\mathcal{B[(0,1)]}$ such that $$\lvert{\mu}\rvert(A) \neq \lvert\mu(A)\rvert $$
Where $\lvert{\mu}\rvert(A)$ is the variation of $\mu$.
My teaching assitant said that we could use "no partitions at all" to compute the variation, but then I'm not sure how to use such definition. I'd appreciate any help on this bad day for math of mine.
This question is wrong. $|\mu| (A)=\sup \{\sum |\mu (A_i)|: A =\bigcup_j A_j, A_j\cap A_k=\emptyset \, \text {for}\, j \neq k\}$. Since $|\mu (A_i)|=|(1+i)| \lambda (A_i)$ we see that $|\mu| (A)=\sqrt 2 \lambda (A)$. Also $|\mu (A)|=|(1+i)| \lambda (A)=\sqrt 2 \lambda (A)$. So $|\mu|(A)=|\mu (A)|$ for every set $A$.
[You can also observe first that $|\mu(A)|=\sqrt2 \lambda (A)$ and then conclude from definition of $|\mu|$ that $|\mu| (A)=\sqrt 2 \lambda (A)$].