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{\bf Question}

Give an outline of the development necessary for a definition of
the Lebesgue intergral of measurable function $f:{\bf R}^n \to
{\bf R}.$  Any theorems concerning measurable functions should be
stated but not proved.

Show that if $f$ and $g$ are two integrable functions then the
function $min(f,g)$ is integratable, and that $$f \, min(f,g) \leq
min \left(\int f, \int g\right).$$ Discuss the case when equality
occurs.

\vspace{.25in}

{\bf Answer}


We first consider the so called simple function.  A function $f:
\Omega \rightarrow R^*$ is called a simple function if it can be
expressed in the form \begin{eqnarray}f(x) &amp; = &amp;  \sum_{i=1}^n c_i
X_{Ei}(x)\end{eqnarray} where $e_i \epsilon R^*$, $\{E_1, E_2,
\ldots E_n\}$ is a partition of $Omega$ into disjoint measurable
sets, and $X_A(x)$ is the characteristic function of the set $A$.

We then introduce the concept of a measurable function.. A
function $f; \Omega \to R^*$ is said to be measurable if and onlt
if for all $c \epsilon R^*$, $\{ x| f(x) \leq c\}$ is a measurable
set.

We the prove the fundemental result stating that any
non-negitative measurable function can be expressed as the limit
if monotonic increasing  sequence of simple functions of the form
(1) by $$\int f = \sum_{i=1}^n c_i \cdot m(E_i),$$ proving htat
$\ds \int f$ is independent of the representation of $f$ in the
form (1).

We then define the integral of a non-negative measurable function
$f$ by expressing $f$ as \begin{eqnarray} f = \lim_{n\to
\infty}f_n \end{eqnarray} where $\{ f_n\}$ is an increasing
sequence of simple functions, and by defining $$\int f =
\lim_{n\to \infty} \int f_n,$$ proving also that $\ds \int f$
independent of the representation of $f$ in the form (2)

We extend the definition to measurable functions which are
positive or negative by adding the functions $f_+, \, f_-:$
\begin{eqnarray*} f_+(x) &amp; = &amp;  \left\{ \begin{array}{rcl} f(x)  &amp; {\rm if} &amp; f(x) &gt;
0 \\ 0 &amp; {\rm if} &amp; f(x) \leq 0 \end{array} \right. \\ &amp; &amp; \\
F_-(x) &amp; = &amp; \left\{ \begin{array}{rcl} -f(x) &amp; {\rm if} &amp; f(x) &lt;
0 \\ 0 &amp; {\rm if} &amp; f(x) \geq 0 \end{array} \right.
\end{eqnarray*} Then for all $x$ we have that $f(x) = f_+(x) -
f_-(x)$.  After proving that for $f$, measurable, $f_+$ and $f_-$
are also measurable, we define $$\int f = \int f_+ - \int f_- $$
whenever R.H.S. is meaningful (i.e.excluding $\infty - \infty$)

${}$

${}$

We now prove that $$ min(f,g) = f - (f-g) _+$$  Suppose $\ds f(x)
\leq g(x)$ then $\ds f(x) - g(x) \leq 0$ So $\ds (f-g) _+(x) = 0$
and $\ds min(f(x),g(x)) = f(x)$ If $\ds f(x) &gt;g(x)$ then $\ds f(x)
- g(x) &gt;0$.  Thus $\ds f(x) - (f-g)_+(x) = g(x) = min
(f(x),g(x)).$

Now the difference of two integrable function is integrable, and
so $$min(f,g) = f- (f-g)_+$$  is integrable.  Also $$\int min(f,g)
= \int f- \int (f-g)_+ \leq \int f$$ since $\ds (f-g)_+ \geq 0$
and so $\int (f-g)_+ \geq 0$

Similarly $$\int min(f,g) \leq \int g,$$and so $$\int min(f,g)
\leq min\left(\int f \, \int g \right).$$

Suppose $\ds min \left( \int f \, \, \int g \right) = \int f$ and
that $\ds \int min(f,g) = \int f$

then $\ds \int (f-g)_+ = 0$.  But $(f-g)_+ \geq 0$ so $(f-g) _+ =
0$ a.e.

Therefore $\ds f(x) \leq g(x)$ a.e.







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