R&C Analysis up to prop 1.25

Rudin's Real and Complex Analysis - Review up to prop 1.25

Main thrust of these sections is to establish Integration of positive functions. All the required machinary for Integration was  established in previous sections.

I felt that section 1.13 and Theorem 1.14 actually should have preceded this section. Only one collollary - The limit of every pointwise convergent sequence of complex measurable functions is measurable is used before this section.

Section 1.13 defines upper limit of sequence $(a_n)$ - called $\beta$ as $$\beta=lim sup_{n \rightarrow \infty} a_n$$ . Lower limit gets defined as negative of upper limit. From real analysis, if $(a_n)$ converges, then both upper and lower limits will equal to limit of sequence.

Based on these definitions, sup and inf of sequence of functions are defined.
$(sup_{n}f_n)(x) = sup_n(f_n(x))$
$(lim sup_{n \rightarrow \infty} f_n)(x) = lim sup_{n \rightarrow \infty}(f_n(x))$

And if
$f(x)= lim_{n \rightarrow \infty} f_n(x)$
the limit being assumed to exist at every point $x \in X$, then $f$ is called pointwise limit of sequence.

Then Theorem 1.14 asserts for $f_n:X \rightarrow [-\infty,\infty]$ is measurable function for $n=1,2,3,\cdots$ and
$g = sup_{n \geq 1}f_n, h = lim sup_{n \rightarrow \infty} f_n$
then $g$ and $h$ are measurable.


Proof is simple. Both the corollaries play a critical role in Integration theory.
(a) The limit of every pointwise convergent sequence of complex measurable functions is measurable.
(b) If $f,g$ are measurable with range in $[-\infty,\infty]$, then so are $max{f,g}$ and $min{f,g}$ and in particular, this is true of the functions
$f^{+} = max\{f,0\}$
$f^{-} = min\{f,0\}$
Proof is simple.

Integration of Positive Functions:
Definition starts off the proceedings for a given $R$ - $\sigma$ algebra and $X$ a measurable space. First order of business to  define Integral of simple functions as this leads to Integral of the given function to which simple functions converge pointwise. Definition pops out very nicely from definition of simple function.
If $s:X \rightarrow [0,\infty]$ is measurable simple function of the form
$s = \sum_{i=1}^\infty \alpha_i \chi_{A_i}$ where $\alpha_1,\cdots,\alpha_n$ are distinct values of $s$ and if $E \in R$, simple function integral is defined as
$\int_E s d\mu = \sum_{i=1}^\infty \alpha_i \mu(A_i \cap E)$.

Lebesgue Integral:
Since multiple simple functions may approximate a given function, its integral is defined as
$\int_E f d\mu = sup \int_E s d\mu$.
The sup taken over all measurable simple functions $s$ such that $0 \leq s \leq f$.

Next all the nice properties of Lebesgue Integral are listed as propostions. Functions and sets are assumed to be measurable.

(a) If $0 \leq f \leq g$ then $\int_E f d\mu \leq \int_E g d\mu$
(b) If $A \subset B$ and $f \geq 0$ then $\int_A f d\mu \leq \int_b f d\mu$.
(c) If $f \geq 0$ and $c$ is constant, $0 \leq c \lt \infty$, then
$\int_E cf d\mu = c \int_E f d\mu$
(d) If $f(x)=0$ for all $x \in E$, then $\int_E f d\mu = 0$ even if $\mu(E) = \infty$.
(e) If $\mu(E)=0$, then $\int_E f d\mu = 0$ even if $f(x) = \infty$ for every $x \in X$.
(f) If $f \geq 0$, then $\int_E f d\mu = \int_X \chi_E f d\mu$.
Last result is nice as it allows to restrict our definition of integration to integrals over all of $X$, without loosing
generality.