Vector valued measure concept from Rudin (Functional Analysis). I've a question, from rudyn functional analysis, by studying the spectral theorem i've seen the concept of "vector valued measures", i haven't seen in the book an explanation of what that means and also the book lacks of examples, which doesn't help in understanding some concepts since the book presents a lots of theorem. My question is if there's a very small introduction that explains what a vector valued measure is and how it is used in the context i mentioned.
 A: If you are interested only in the spectral theorem, then you're working in a Hilbert space, and everything concerning the measures can be reduced to complex measures by considering $\mu_{x,y}(S)=(E(S)x,y)$, where $E$ is the spectral measure. In fact, on a complex Hilbert space, the study reduces to the positive measures $\mu_{x}(S)=(E(S)x,x)$ through the polarization identity. Then, for example, if you have a bounded Borel measurable function $F$,
$$
                \int F(\lambda)d(E(\lambda)x,y)=(Ax,y)
$$
for a unique bounded operator $A$, and you can use $A$ to define $\int F(\lambda)dE(\lambda)$, giving the suggestive definition
$$
              \int F(\lambda)d\mu_{x,y}(\lambda)=\int F(\lambda)d(E(\lambda)x,y) = \left(\int F(\lambda)dE(\lambda)x,y\right)
$$
That's a most efficient way to define the integral objects and, basically, any other definition that reduces to the same weak objects must yield the same integral when dealing with the same objects such as bounded Borel functions $F$. This is one of the reasons the subject is called Functional Analysis.
Uniqueness of the Riesz Representation for functionals becomes a fundamental way to transfer weak convergence properties to strong convergence properties. For example, if $F$ is a simple function $F=\sum_{n}a_n \chi_{S_n}$, then the above definition gives
\begin{align}
         \int F(\lambda)d(E(\lambda)x,y) & = \sum_{n}a_n (E(S_n)x,y) \\
       & = \left(\sum_n a_n E(S_n)x,y\right)
\end{align}
By uniqueness of representation, the operator $\int F(\lambda)dE(\lambda)$ as defined in the previous paragraph becomes
$$
                  \int F(\lambda)dE(\lambda) = \sum_n a_n E(S_n).
$$
So this definition gives you what you expect for simple functions. The left side is an operator. To apply this operator to a vector,
$$
       \left(\int F(\lambda)dE(\lambda)\right)x = \sum_n a_n E(S_n)x
$$
It is convention to the write the left side as $\int F(\lambda)dE(\lambda)x$. For such a simple function, if the $S_n$ are disjoint, then the vectors $E(S_n)x$ are mutually orthogonal, which gives
$$
           \left\|\int F dE x \right\|^2 = \sum_n |a_n|^2\|E(S_n)x\|^2
           = \int |F(\lambda)|^2 d\|E(\lambda)x\|^2
$$
Then you can bootstrap to general bounded Borel functions through limits of simple Borel functions. Because $E(S)$ is an orthogonal projection
$$
            \mu_x(S)=(E(S)x,x) = (x,E(S)x) = \|E(S)x\|^2.
$$
You also obtain
$$
         \left(\int fdEx,y\right)= \int fd(Ex,y) = \left(x,\int \overline{f}dEy\right), \\
            \left(\int fdE\right)^{\star} = \int \overline{f}dE.
$$
