A function is integrable if and only if the nets of lower sum and upper sum converge to the same number.

Question

Definition 1

If $\mathscr{P}$ is the set of all partition of a rectangle $Q$ of $\Bbb{R}^n$ then we say that $$ P_1\preccurlyeq P_2\,\Leftrightarrow\,\text{any point of} \,P_1\,\text{is a point of}\,P_2 $$ for any $P_1,P_2\in\mathscr{P}$.

Lemma 2

The set $\mathscr{P}$ equipped with the relation $\preccurlyeq$ is a directed set.

Proof. Clearly the relations $\preccurlyeq$ is reflexive and transitive and so we observe that for any $P_1,P_2\in\mathscr{P}$ there exist $(P_1\cup P_2)\in\mathscr{P}$ such that $P_1,P_2\preccurlyeq(P_1\cup P_2)$. So we conclude that $(\mathscr{P},\preccurlyeq)$ is a directed set.

Definition 3

If $\mathscr{P}$ is the set of all partition a rectangle $Q$ of $\Bbb{R}^n$ then for any function $f:Q\rightarrow\Bbb{R}$ we define the nets $\lambda:\mathscr{P}\rightarrow\Bbb{R}$ and $\upsilon:\mathscr{P}\rightarrow\Bbb{R}$ through the condiction $$\lambda(P):=L(f,P)$$ for any $P\in\mathscr{P}$ and $$\upsilon(P):=U(f,P)$$ for any $P\in\mathscr{P}$.

Definition 4

If $Q$ a rectangle is a rectangle of $\Bbb{R}^n$ and if $f:Q\rightarrow\Bbb{R}$ then as $P$ ranges over all partitions of $Q$ we define $$ \underline{\int}_Q f:=\sup\{L(f,P)\}\,\,\,\text{and}\,\,\,\overline{\int}_Q f:=\inf\{U(f,P)\} $$ and we call it lower integral and upper integral so that we say that $f$ is integrable over $Q$ if and only if these two numers are equal.

Lemma 5

If $Q$ is a rectangle of $\Bbb{R}^n$ and if $f:Q\rightarrow\Bbb{R}$ is a function then it is integrable over $Q$ is and only if for any $\epsilon>0$ there exist a partition $P$ of $Q$ such that $U(f,P)-L(f,P)<\epsilon$.

Theorem 6

If $Q$ is a rectangle of $\Bbb{R}^n$ and if $f:Q\rightarrow\Bbb{R}$ is a function then it is integrable over $Q$ if and only if the nets $\lambda$ and $\upsilon$ converge to the same number and moreover this is the integral of the function $f$ over $Q$.

Unfortunately I can't prove the last theorem. So could someone help me, please?

Answer 1

Theorem

If $Q$ is a rectangle of $\Bbb{R}^n$ and if $f:Q\rightarrow\Bbb{R}$ is a function then it is integrable over $Q$ if and only if the nets $\lambda$ and $\upsilon$ converge to the same number and moreover this is the integral of the function $f$ over $Q$.

Proof. So we suppose that the nets $\lambda$ and $\upsilon$ converge to the number $\xi$ so that for any $\epsilon>0$ there exist partitions $P_\lambda,P_\upsilon\in\mathscr{P}$ such that $\xi-\frac{\epsilon}2<L(f,P)$ for any $P\in\mathscr{P}$ such that $P\succcurlyeq P_\lambda$ and $U(f,P)<\xi+\frac{\epsilon}2$ for any $P\in\mathscr{P}$ such that $P\succcurlyeq P_\upsilon$ and so for $P_\xi\in\mathscr{P}$ such that $P_\lambda,P_\upsilon\preceq P_\xi$ (remember tha $\mathscr{P}$ is a directed set) it follows that

$$ \xi-\frac{\epsilon}2<L(f,P_\xi)\le U(f,P_\xi)<\xi+\frac{\epsilon}2 $$ and so $$ U(f,P_\xi)-L(f,P_\xi)<\big(\xi+\frac{\epsilon}2)-(\xi-\frac{\epsilon}2)=\epsilon $$ so that the function $f$ is integrable over $Q$ by lemma 5. Now by definition $4$ and by the second to last inequality we know that $$ \xi-\frac{\epsilon}2<\int_Q f<\xi+\frac{\epsilon}2 $$ and so $$ |\int_Q f-\xi|<\frac{\epsilon}2<\epsilon $$ so that by the arbitrariness of $\epsilon>0$ we conclude that $\xi=\int_Q f$.

Conversely we suppose that the function $f$ is integrable over $Q$. So for any $\epsilon>0$ by the properties of the supremum and infimum there must exist $P_{_{L}},P_{_{U}}\in\mathscr{P}$ such that $$ \int_Qf-\epsilon<L(f,P_{_{L}})\,\,\,\text{and}\,\,\,U(f,P_{_{U}})<\int_Q f+\epsilon $$ and so for $P_0\in\mathscr{P}$ such that $P_{_{U}},P_{_{L}}\preccurlyeq P_0$ (remember that $\mathscr{P}$ is a directed set) it follows that the two inequalities hold for any $P\in\mathscr{P}$ such that $P\succcurlyeq P_0$ because if $P\succcurlyeq P_0\succcurlyeq P_{_{L}},P_{_{U}}$ then $P$ is a refiniment of $P_0$ and $P_0$ is a refiniment of $P_L$ and $P_U$ thus $$ L(f,P_{_{L}})\le L(f,P_0)\le L(f,P)\,\,\,\text{and}\,\,\,U(f,P)\le U(f,P_0)\le U(f,P_{_{U}}) $$ so that if we remember that $$ L(f,P)\le\int_Q f<\int_Q f+\epsilon\,\,\,\text{and}\,\,\,\int_Q f-\epsilon<\int_Q f\le U(f,P) $$ for any $P\in\mathscr{P}$ then $$ \Big|\int_Q f-L(f,P)\Big|<\epsilon\,\,\,\text{and}\,\,\,\Big|U(f,P)-\int_Q f\Big|<\epsilon $$ for any $P\in\mathscr{P}$ such that $P\succcurlyeq P_0$. So we conclude that the nets $\lambda$ and $\upsilon$ converge to $\int_Q f$.