# integral related a spectral family

#### nassima2231

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
Hey nassima2231.

What are the properties of a spectral family?

#### nassima2231

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.)