Theorem: Vitali-Carathéodory Theorem

Let $\struct {X, \tau}$ be a locally compact Hausdorff space.
Let $\MM$ be a $\sigma$-algebra over $X$ which contains the Borel $\sigma$-algebra generated by $\tau$.
Let $\mu$ be a Radon measure on $\MM$.
Let $f \in \map {\LL^1} \mu$, where $\map {\LL^1} \mu$ denotes the (real) Lebesgue 1-space of $\mu$.
For all $\epsilon \in \R_{>0}$, there exists some $\tuple {u, v} \in \paren {X^\R}^2$ such that:
:$u$ is upper semicontinuous and bounded above
:$v$ is lower semicontinuous and bounded below
:$u \le f \le v$
and :
:$\ds \int_X \paren {v - u} \rd \mu < \epsilon$.

Proof:
Let:
:$\forall x \in X: \map f x \ge 0$
and:
:$\exists x \in X: \map f x \ne 0$
By Measurable Function is Pointwise Limit of Simple Functions, there exists a sequence:
:$\sequence {s_n} \in \paren {\map \EE \MM}^\N$
where $\map \EE  \MM$ denotes the space of simple functions on $\struct {X, \MM}$.
By Pointwise Difference of Simple Functions is Simple Function, the differences of consecutive terms in a sequence of simple functions are simple functions.
By Limit of Sequence is Sum of Difference of Consecutive Terms, there exists a sequence $\sequence {t_n}$ of simple functions such that:
:$\ds f = \sum_{i \mathop = 1}^\infty t_n$
By the definition of simple functions, each simple function is a finite linear combination of characteristic functions.
Thus there exists some $\tuple {\sequence {E_i}, \sequence {c_i} } \in \MM^\N \times \R_{>0}^\N$ such that:
:$\ds f = \sum_{i \mathop = 1}^\infty c_i \chi_{E_i}$
Now:
{{begin-eqn}}
{{eqn | l = \sum_{i \mathop = 1}^\infty c_i \map \mu {E_i}
      | r = \sum_{i \mathop = 1}^\infty \int_X c_i \chi_{E_i} \rd \mu
      | c = Integral of Characteristic Function
}}
{{eqn | r = \int_X \sum_{i \mathop = 1}^\infty c_i \chi_{E_i} \rd \mu
      | c = Monotone Convergence Theorem (Measure Theory)
}}
{{eqn | r = \int_X f \rd \mu
}}
{{eqn | o = <
      | r = \infty
      | c = as $f \in \map {\LL^1} \mu$
}}
{{end-eqn}}
That is, the series:
:$\ds \sum_{i \mathop = 1}^\infty c_i \map \mu {E_i}$
converges. 
Denote by $\powerset X$ the power set of X.
By the definition of Radon measure, for all $\epsilon \in \R_{>0}$, there exists some $\tuple {\sequence {K_i}, \sequence {V_i} } \in \paren {\paren {\powerset X}^\N}^2$ such that for all $i \in \N$:
:$K_i$ is compact
:$V_i$ is open
:$K_i \subset E_i\subset V_i$
and
:$c_i \map \mu {V_i - K_i} < 2^{-\paren {i + 1} } \epsilon$
By Characteristic Function of Open Set is Lower Semicontinuous:
:for all $i \in \N$, $\chi_{V_i}$ is lower semicontinuous. 
By Constant Multiple of Lower Semicontinuous Function is Lower Semicontinuous:
:for all $i \in \N$, $c_i \chi_{V_i}$ is lower semicontinuous.
Define:
:$\ds v = \sum_{i \mathop = 1}^\infty c_i \chi_{V_i}$
By Series of Lower Semicontinuous Functions is Lower Semicontinuous:
:$v$ is lower semicontinuous.
By Characteristic Function of Compact Set is Upper Semicontinuous:
:for all $i \in \N$, $\chi_{K_i}$ is upper semicontinuous.
By Constant Multiple of Upper Semicontinuous Function is Upper Semicontinuous:
:for all $i \in \N$, $c_i \chi_{K_i}$ is upper semicontinuous.
By definition of convergent series, for all there exists some $N \in \N$ such that:
:$\ds \sum_{i \mathop = N + 1}^\infty c_i \map \mu {E_i} < \frac \epsilon 2$
Define:
:$\ds u = \sum_{i \mathop = 1}^N c_i \chi_{K_i}$
By Finite Sum of Upper Semicontinuous Functions is Upper Semicontinuous, $u$ is upper semicontinuous.
Now, for all $i \in \N$:
:$\chi_{K_i} \le \chi_{E_i} \le \chi_{V_i}$
So:
:$u \le f \le v$
Now:
{{begin-eqn}}
{{eqn | l = \int_X \paren {v - u} \rd \mu
      | r = \int_X \paren {\sum_{i \mathop = 1}^N c_i \paren {\chi_{V_i} - \chi_{K_i} } + \sum_{i \mathop = N + 1}^\infty c_i \chi_{V_i} } \rd \mu
}}
{{eqn | o = \le
      | r = \int_X \paren {\sum_{i \mathop = 1}^\infty c_i \paren {\chi_{V_i} - \chi_{K_i} } + \sum_{i \mathop = N + 1}^\infty c_i \chi_{E_i} } \rd \mu
}}
{{eqn | r = \sum_{i \mathop = 1}^\infty \int_X c_i \paren {\chi_{V_i} - \chi_{K_i} } \rd \mu + \sum_{i \mathop = N + 1}^\infty \int_X c_i \chi_{E_i} \rd \mu
      | c = Monotone Convergence Theorem (Measure Theory)
}}
{{eqn | r = \sum_{i \mathop = 1}^\infty c_i \map \mu {V_i - K_i} + \sum_{i \mathop = 1}^\infty \map \mu {E_i}
      | c = Integral of Characteristic Function: Corollary
}}
{{eqn | o = <
      | r = \sum_{i \mathop = 1}^\infty \frac \epsilon {2^{i + 1} } + \frac \epsilon 2
}}
{{eqn | r = \epsilon
}}
{{end-eqn}}
{{qed}}
{{Namedfor|Giuseppe Vitali|cat = Vitali|name2 = Constantin Carathéodory|cat2 = Carathéodory}}
Category:Real Analysis
Category:Hausdorff Spaces
Category:Lower Semicontinuity
Category:Upper Semicontinuity
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