integral related a spectral family

Dec 2014
3
0
Oran
Hello,
how to define an integral from a spectral family?
for Riemann integral, we fix an epsilon, we use upper sums, lower sums, a partition, etc.... then we proof that this integral is independent for the choice of this partition and this epsilon , but here what is the main idea? how is it constructed? is there notion of "it's well defined" ?please help me.
 

chiro

MHF Helper
Sep 2012
6,608
1,263
Australia
Hey nassima2231.

What are the properties of a spectral family?
 
Dec 2014
3
0
Oran
Hy, (yesterday I couldn't see your reply, because there was a problem in my computer, i'm sorry),
the property of a spectral measure :
Let Y be a locally compact, $\sigma$-compact, topological space. A spactral measure, on Y is a map E : B(Y) --> L(H) with the following properties:
1) E(M) is an orthogonal projection for each M in B(Y); E(\emptyset)=0, E(Y)=I.
2)E(M_1 \cap M_2)=E(M_1)E(M_2) for all M_1,M_2 in B(Y).
3)E(M_1 \cup M_2)=E(M_1)+E(M_2) for all M_1,M_2 in B(Y) wih M_1\cap M_2=emptyset.
4)For every x in H, E_x,x : M ---> <E(M)x,x> is a measure on Y.
(i'm new in this website, tell me why can i use tex, if it's possible please.)