Integrable Functions with Equal Integrals on Sub-Sigma-Algebra are A.E. Equal

== Theorem ==
Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $\GG$ be a sub-$\sigma$-algebra of $\Sigma$.
Let $f, g: X \to \overline \R$ be $\GG$-integrable functions.
Suppose that, for all $G \in \GG$:
:$\ds \int_G f \rd \mu = \int_G g \rd \mu$
Then $f = g$ $\mu$-almost everywhere.
== Proof ==
Define: 
:$A = \set {x \in X : \map f x \in \R}$
and:
:$B = \set {x \in X : \map g x \in \R}$ 
From Set of Points for which Measurable Function is Real-Valued is Measurable, we have that: 
:$A$ and $B$ are $\GG$-measurable. 
From Sigma-Algebra Closed under Countable Intersection, we have that: 
:$A \cap B$ is $\GG$-measurable.
Consider the characteristic function $\chi_{A \cap B}$. 
From Characteristic Function Measurable iff Set Measurable, we have that: 
:$\chi_{A \cap B}$ is $\GG$-measurable.
From De Morgan's Laws: Complement of Intersection, we have
:$X \setminus \paren {A \cap B} = \paren {X \setminus A} \cup \paren {X \setminus B}$ 
We have that: 
:$X \setminus A = \set {x \in X : \size {\map f x} = +\infty}$
and:
:$X \setminus B = \set {x \in X : \size {\map f x} = +\infty}$ 
From Integrable Function is A.E. Real-Valued, we have that: 
:$X \setminus A$ and $X \setminus B$ are $\mu$-null. 
From Null Sets Closed under Countable Union, we then have that: 
:$\paren {X \setminus A} \cup \paren {X \setminus B}$ is $\mu$-null.
Let $G \in \GG$. 
Note that if $x \in X$ is such that: 
:$\map f x \map {\chi_G} x \map {\chi_{A \cap B} } x \ne \map f x \map {\chi_G} x$ 
Then: 
:$\map {\chi_{A \cap B} } x = 0$
so:
:$x \in X \setminus \paren {A \cap B}$
so:
:$x \in \paren {X \setminus A} \cup \paren {X \setminus B}$
Giving:
:$f \cdot \chi_G \cdot \chi_{A \cap B} = f \cdot \chi_G$ $\mu$-almost everywhere.
Swapping $f$ for $g$, we also obtain:
:$g \cdot \chi_G \cdot \chi_{A \cap B} = g \cdot \chi_G$ $\mu$-almost everywhere.
Additionally, from Pointwise Product of Measurable Functions is Measurable, we have that: 
:$f \cdot \chi_G \cdot \chi_{A \cap B}$ and $g \cdot \chi_G \cdot \chi_{A \cap B}$ are $\GG$-measurable for each $G \in \GG$. 
By hypothesis, we have that: 
:$\ds \int_G f \rd \mu = \int_G g \rd \mu$
for each $G \in \GG$, so:
:$\ds \int f \cdot \chi_G \rd \mu = \int g \cdot \chi_G \rd \mu$
Then from A.E. Equal Positive Measurable Functions have Equal Integrals, we have: 
:$\ds \int f \cdot \chi_G \cdot \chi_{A \cap B} \rd \mu = \int g \cdot \chi_G \cdot \chi_{A \cap B} \rd \mu$
so:
:$\ds \int_G f \cdot \chi_{A \cap B} \rd \mu = \int_G g \cdot \chi_{A \cap B} \rd \mu$
for each $G \in \GG$. 
That is: 
:$\ds \int_G \paren {f \cdot \chi_{A \cap B} - g \cdot \chi_{A \cap B} } \rd \mu = 0$ 
for each $G \in \GG$. 
Then, from Integrable Function with Zero Integral on Sub-Sigma-Algebra is A.E. Zero, we have that: 
:$f \cdot \chi_{A \cap B} - g \cdot \chi_{A \cap B} = 0$ $\mu$-almost everywhere.
That is: 
:$f \cdot \chi_{A \cap B} = g \cdot \chi_{A \cap B}$ $\mu$-almost everywhere.
So, if $x \in X$ is such that $\map f x \ne \map g x$, we must have: 
:$\map {\chi_{A \cap B} } x = 0$
That is: 
:$x \in X \setminus \paren {A \cap B}$ 
and:
:$X \setminus \paren {A \cap B}$ is a $\mu$-null set.
So:
:$f = g$ $\mu$-almost everywhere.
== Note ==
It may be tempting to write that: 
:$\ds \int_G \paren {f - g} \rd \mu = 0$
for each $G \in \GG$, and conclude immediately from Integrable Function with Zero Integral on Sub-Sigma-Algebra is A.E. Zero.
However, the pointwise difference $f - g$ may not be well-defined. 
For example, if there is a point $x \in X$ such that $\map f x = \map g x = +\infty$. 
We avoid this technicality by introducing the sets $A$ and $B$, as subtraction is well-understood on $A \cap B$. 
We write $f \cdot \chi_{A \cap B} - g \cdot \chi_{A \cap B}$ instead of $\paren {f - g} \cdot \chi_{A \cap B}$ for this reason. 
== Also see ==
* Measurable Functions with Equal Integrals on Sub-Sigma-Algebra are A.E. Equal
== Sources ==
* : $10.14 \ \text{(i)}$
Category:Measure Theory