prove that : lambda (A+B)= Lambda A +Lambda B
Answers
Answer:idk sorry bro
Step-by-step explanation:
Answer:
Prove that for the Lebesgue-measure λ, the inequality
λ(A+B)≥λ(A)+λ(B)
holds.
So, this task is divided in some smaller tasks, and I'm supposed to begin with the following one:
Show that the inequality holds for half-open intervals
A=[a1,b1) and B=[a2,b2)
with a1,a2,b1,b2∈R, b1>a1 and b2>a2.
I tried to do a case analysis, taking a look at A⊂B, A∩B=∅ and A∩B≠∅. Yet, I only came to the conclusion that, no matter what case I try to prove, that there is always only the equality between A and B.
So, we know that A+B=[a1,b1)+[a2,b2)=[a1+a2,b1+b2). This leads to
λ(A+B)=λ([a1+a2,b1+b2))=(b1+b2)−(a2+a1)=b1+b2−a2−a1=b1−a1+b2−a2=λ([a1,b1))+λ([a2,b2))=λ(A)+λ(B).
Of course this doesn't contradict the statement above, but I guess there is more to show here. We are allowed to use the usual properties of a measure.
Edit:
So, basically, I couldn't find an example where the equality didn't hold. I guess this is exactly sub-task wats me to find? We later work with A and B being compact, so I might get a different result, but I'm not sure.
EditEdit:
It turned out that my solution is perfectly fine, I simply refused to believe that the equation would hold for every interval - but it does.