Question

Let (E,E, u) be a measure space. Let De E (a) Define (A) = u(AnD), A E E. Show that v is a measure on (E,E); it is called the trace of u on D. (b) Let D be the trace of E on D (see last homework). Define v(A) is a measure on (D, D); it is called the restriction of u to D (A) for A e D. Show that v

Answer 1

E,E,u
is a measure space and
DES
.

a) Define .

A (An D), A E E

Non negativity:
(A (AnD)0
, hence
V(A)>
.

Null empty set:

nD = /4( )

Countable additivity:

(U1 (U D)(U(AnD) n-1 n=1 n=1

.

-ΣμΑn D) -ΣνAΑ) n-1 n=1

Hence $\nu$ satisfies all the properties of a measure. Therefore it is a measure.

b) $\mathcal{D}$ is the trace of $\mathcal{E}$, that is, $\mathcal{D} = \{ A \cap D : A \in \mathcal{E} \}$ .

Now if $A \in \mathcal{D}$ , then .

ANDA

Hence $\nu(A) = \mu(A) = \mu (A \cap D)$ .

In problem a we have shown that  
(AND)
is a measure. By restriction it D, we get that $\nu$ is a measure on $(D, \mathcal{D} )$ .