Spectral integral: reference request Consider the following fragment from Takesaki's book "Theory of operator algebra I":

In this proof, we encounter expressions like
$$x = \int_{-\|x\|}^{\|x\|}\lambda de(\lambda).$$
Can someone explain me how to understand this integral or give me an 'accessible' reference where I can read more about these kinds of integrals?  Is this related to the "Borel functional calculus"?
 A: This is a result of the spectral theorem for self adjoint, bounded operators on a Hilbert space.  The theorem which can be found in Reed and Simons for example provides a way of constructing an operator out of an integral over the spectrum of the operator with respect to a special kind of measure called a spectral measure.
In the case of self adjoint $x$ we know $\sigma(x)\subset \mathbb{R}$ and moreover, the spectral radius is equal to $\|x\|$.
The $de(\lambda)$ is the spectral measure defined on $\sigma(x)$ and it allows you to integrate continuous functions or even measurable functions with respect to this measure to define the analogue of the given function on the operator $x$.  For example, if $f(\lambda)=e^\lambda$ defined on the spectrum of $x$, then the spectral theorem says that
$$f(x)=\int_{\sigma(x)}f(\lambda)de(\lambda)$$
So you can give meaning to the exponential of an operator (in the case of bounded operators this machinery is not necessary since you can represent it as a power series in the operator norm).
In the case where  $f(\lambda)=\lambda$ i.e the $id_{\sigma(x)}$ then the theorem yields
$$x=\int_{\sigma(x)}\lambda de(\lambda)$$
But remember if $x$ is self adjoint then the spectral radius is equal to $\|x\|$ and $\sigma(x)\subset \mathbb{R}$ we have $\sigma(x)\subset [-\|x\|,\|x\|]$ (if $\sigma(x)\subsetneq [-\|x\|,\|x\|]$ you can extend the integration to the endpoints by declaring that the spectral measures are exactly zero in the compliment) and hence
$$x=\int_{-\|x\|}^{\|x\|}\lambda de(\lambda)$$
