I am learning spectral integration for my summer. I am stuck at a point.
Having got hold of a spectral measure, we define the spectral integral of a simple function as usual and then approximate any bounded measurable function in uniform norm by a sequence of simple functions and then the integral of the bounded measurable function is defined as the limit of the sequence of integrals of the approximating simple functions (that this sequence converges can be shown).
Then for the unbounded functions, we approximate pointwise by bounded functions specifically if $f$ is an unbounded measurable function on a measure space $(X, \mathcal{A}, \mu)$ then define
$$ M_{n} = \{t \in X \,| \quad |f(t)| \leq n \} $$
and define
$$ f_{n} = f\chi_{M_{n}}.$$
Then we can define the integral of $f$ at a vector $x$ as
$$ I(f)x = \lim_{n}I(f_{n})x $$
for all those $x$ for which the above limit exists.
I understand that if the function is bounded, then the integral operator so obtained is bounded and its norm is equal to the sup-norm of the function.
However, I cannot see that if the integral of the function is bounded, then the function should be essentially bounded. Could some one give me any leads?
I am following Schmudgen's "Unbounded Self-Adjoint Operators on Hilbert Space", but I am not able to follow the proof he gives. Also, the proof seems a little too complicated.