First a few definitions.
A measurable set is called null for signed or complex measure if each of its measurable subset has zero -value. This is equivalent to the condition that the value of total variation of this measurable set is 0.
For two measure and [/tex]\lambda_2[/tex], we say [tex]\lambda_1\perp\lambda_2[tex] if there are two measurable sets E and F such that and .
For two signed measure and , we say if there are two measurable sets E and F such that and E is null for and F is null for . The definition of mutual sigularity for measures agrees with that for signed measures.
Now I have found two kinds of definitions of mutual singularity for complex measures:
For two complex measures and , they are said to be mutually singular if
Definition one: where and are real and imaginary parts of , respectively, and are real and imaginary parts of , respectively. Note that are all (finite) signed measures.
Definition two: where denotes total variation. Note that and are both (finite) measures. This definition can be proved to be equivalent to: there are two measurable sets E and F such that and E is null for and F is null for .
I'm trying to prove that these two definitions are equivalent. I have proved that Definitin two can imply Definition one. But I met with difficulties when proving the converse. The follwing is my attempt: For , suppose and make the mutual regularity, i.e., and E is null for and F is null for . For , suppose and make the mutual regularity. Likewise, suppose and for , and for . I wish to find E and F such that E is null for and F is null for to meet the condition in Definition two. I construct , (the complement of E) . Since E is a subset of , E is null for . Since E is a subset of , E is null for , so E is null for . But for F, subadditivity shows that , if I can obtain (and other three terms) is equal to 0, we can arrive at the desired consequence. But is only null for so I have only . I hope I can prove , but Unfortunately, like the amplitude of real part is always smaller than the amplitude of the complex number, the realtion between and is . So the above construction does not work. I can not find other methods to construct the desired E and F and I was stuck here. It's not a homework or exercise and I have thought several days for it. I think I am unable to work it out by myself so I post this question here hoping someone could help me. Any clue or hint is welcomed, thanks!
If you need relevant information such as a specific definition, please tell me and I'll post it (after 15 hours because I have to go to bed now). Thanks!