Suppose $C,E\subset\Bbb R^2$ are bounded s. t. $C\cap E$ is of Lebesgue measure $0$ and let $A\supseteq C\cup E.$ If $f:A\to\Bbb R$ is (Riemann) integrable on $C$ and $E,$ is it necessarily (Riemann) integrable on $C\cap E$?
This question arose when I was revising the theorem about the additivity of an integral w.r.t. to the area of integration:
$\underline{\boldsymbol{\text{ theorem }8.9.}:}$
Suppose $C,E\subset\Bbb R^2$ are bounded s. t. $C\cap E$ is of Lebesgue measure $0$ and that $f:C\cup E\to\Bbb R$ is a function integrable on $C,E,$ and $C\cap E$. Then $f$ is integrable on $C\cup E$ and $$\int_{C\cup E}f=\int_C+\int_Ef.$$
Just in case, I'm going to write down the proof for the context:
Let $A$ be a rectangle containing $C\cup E$ and let's note the extension of $f$ by $0$ from $C\cup E$ to $A$ by $\tilde f.$ We're going to observe the characteristic functions $\chi_C,\chi_E$ and $\chi_{C\cap E}$ of the sets $C,E$ and $C\cap E$ as functions on $A$. Let $f_1=\tilde f\chi_C,f_2=\tilde f\chi_E$ and $f_3=\tilde f\chi_{C\cap E}.$ Then $f_1,f_2$ and $f_3$ are extensions by $0$ of $f_{\mid C},f_{\mid E}$ and $f_{\mid C\cap E}$ on $A,$ respectively. Therefore, $f_1,f_2$ and $f_3$ are integrable on $A.$ Clearly $\chi_{C\cup E}=\chi_C+\chi_E-\chi_{C\cap E}.$ It follows $$\tilde f=\tilde f\chi_{C\cup E}=f_1+f_2-f_3.$$ Hence, $\tilde f$ is integrable on $A$ and it holds $\int_A\tilde f=\int_Af_1+\int_Af_2-\int_Af_3.$ Since $\int_A\tilde f=\int_{C\cup E},\int_Af_1=\int_Cf,\int_Af_2=\int_Ef$ and $\int_Af_3=0,(*)$ the claim follows.
$(*)$ is the result proven shortly before:
$\underline{\boldsymbol{\text{ proposition }8.6}}:$
If $C\subset\Bbb R^2$ is bounded and of Lebesgue measure $0$ and $f:C\to\Bbb R$ is integrable on $C,$ (hence, bounded) then $\int_Cf=0.$
Question:
Would the same claim hold without the assumption that $f$ is (Riemann) integrable on $C\cap E,$ that is, is that assumption redundant if other assumptions of the theorem remain?
I thought of first partitioning $C$ and $E$ separately and then $C\cap E$ as the subset of which and then find the common refinement of $P_C$ and $P_{C\cap E}$ and of $P_E$ and $P_{C\cap E}$ and comparing Darboux sums, however I haven't come up with anything. An idea of taking some rectangle $B$ containing $C\cap E$ and an arbitrarily small stripe around $\partial B$ was also floating around…
EDIT (for clarification)
That $f$ is (Riemann) integrable on $C\cap E$ is part of the assumption of the above stated theorem. I was wondering whether that assumption was redundant if $f$ might necessarily be (Riemann) integrable on $C\cap E,$ so my question boils down to:
If a function $f$ is (Riemann) integrable on two bounded sets $C,E\subset\Bbb R^2,$ and their intersection is of Lebesgue measure $0,$ is it necessarily (Riemann) integrable on the intersection, too?
Now that I wrote this, if we forget the assumption on the intersection being of Lebesgue measure $0,$ would that affect integrability?
EDIT ( $6^{\mathrm{th}}$ April 2022):
I would like to include the lemma:
$\underline{\boldsymbol{\text{ lemma }8.8:}}$
Suppose $C\subseteq\Bbb R^2$ is a set of Jordan measure $0$ and $f: C\to\Bbb R$ is bounded. Then $f$ is integrable and $\int_Cf=0.$
Maybe now my question can boil down to:
If $C,E\subset\Bbb R^2$ are bounded, $C\cap E$ is of Lebesgue measure $0,$ and $f:C\cup E\to\Bbb R$ is integrable on both $C$ and $E,$ is it possible that $C\cap E$ has no Jordan measure, that is, $\chi_{C\cap E}$ isn't integrable on any rectangle $A\supset C\cap E$?
Because if $C\cap E$ is of Lebesgue measure $0$ and has Jordan measure, its Jordan measure is necessarily $0$ and $f$ is integrable on $C\cap E$ by the lemma 8.8.
Best Answer
We have the following proposition (regardless of the measure of $C \cap E$).
The proof will rely on the following lemma (which you should be able to prove in a straightforward way by showing that the min and max preserve continuity.)
Proof of proposition.
Let $f^+(x) = \max (f(x),0)$ and $f^-(x) = \max (-f(x),0)$. Then $f^+$ and $f^-$ are nonnegative functions such that $f(x) = f^+(x) - f^-(x)$. If $f$ is Riemann integrable on $C$ and $E$, then it follows from the lemma that $f^+$ and $f^-$ are Riemann integrable on $C$ and $E$.
Using the notation $f^+_S := f^+\chi_S$ we have, since $f^+$ is nonnegative,
$$f^+_{C\cap E}(x) = \min(f^+_C(x), f^+_E(x))$$
SInce $f^+$ is Riemann integrable on $C$ and $E$, we have $f^+_C$ and $f^+_E$ Riemann integrable on a rectangle $A$ containing $C$ and $E$ (and, hence, $C\cap E$). By the lemma it follows that $f^+_{C\cap E}$ is Riemann integrable on A and, by definition, $f^+$is Riemann integrable on $C \cap E$.
By the same argument we can show that $f^-$ is Riemann integrable on $C \cap E$ and by integral additivity $f = f^+-f^-$ is Riemann integrable on $C \cap E$.