lemmas on integrals

Question

I tried to prove the following lemmas: if $f,g$ are two bound functions in $[a,b]$, then: $\underline{\int_{a}^{b}}{f(x)dx}+\underline{\int_{a}^{b}}{g(x)dx}\leq\underline{\int_{a}^{b}}{(f+g)(x)dx}$

and let $c$ be a point in $(a,b)$, then:

$\overline{\int_{a}^{b}}{f(x)dx}=\overline{\int_{a}^{c}}{f(x)dx}+\overline{\int_{c}^{b}}{f(x)dx}$

note that $\underline{\int_{a}^{b}}{f(x)dx}$ is the lower Darboux integral of $f(x)$ and $\overline{\int_{a}^{b}}{f(x)dx}$ is the upper Darboux itegral when using Darboux's integral. I tried to prove the first one through the defintion itself but I got stuck when the $m_i=\underset{x\in[x_{i-1},x_i]}{\inf}{f(x)}$ used for the different functions are'nt the same, and when trying to chose the maximum of the two I got stuck with two of that, which doesn't lead me to the right side of the inequality. I would preffer an answer that comes straight from the definition.

In the second lemma I've managed to prove that the right side is greater or equal to the left side, so all I have left is to prove that: $\overline{\int_{a}^{b}}{f(x)dx}\geq\overline{\int_{a}^{c}}{f(x)dx}+\overline{\int_{c}^{b}}{f(x)dx}$

thanks!

Answer 1

For the first lemma, since $\inf f(x) + \inf g(x) \leqslant f(x) + g(x)$ we have the inequality

$$\inf_{x \in [x_{i-1},x_i]} f(x) + \inf_{x \in [x_{i-1},x_i]} g(x) \leq \inf_{x \in [x_{i-1},x_i]} [f(x) + g(x)],$$

where $[x_{i-1},x_i]$ is a subinterval of some partition $P$.

This implies, for lower Darboux sums, that

$$\begin{equation*}L(P,f) + L(P,g)\leq L(P,f+g) \tag{1}\end{equation*}$$

Proving the desired inequality is a bit tricky, since taking suprema over lower sums directly leads to a dead end.

For example, we see immediately that

$$\sup_{P}[L(P,f) + L(P,g)]\leq \sup_{P}L(P,f+g)=\underline{\int}_a^b[f(x)+g(x)]\, dx \tag{2}$$

and

$$ L(P,f) + L(P,g) \leq \sup_{P}L(P,f) + \sup_{P}L(P,g) = \underline{\int}_a^bf(x)dx + \underline{\int}_a^bg(x)dx\\ \implies \sup_{P}(L(P,f) + L(P,g))\leq \underline{\int}_a^bf(x)dx + \underline{\int}_a^bg(x)dx \tag{3}$$

Unfortunately (2) and (3) do not lead to an ordering of the lower Darboux integrals on the RHS of each inequality.

Instead, we can arrive at the result indirectly. Assume on the contrary that

$$\underline{\int}_a^b [f(x)+g(x)] \, dx < \underline{\int}_a^b f(x) \, dx + \underline{\int}_a^b g(x) \,dx .$$

Rearrranging, we get

$$\underline{\int}_a^b [f(x)+g(x)] \, dx - \underline{\int}_a^b g(x) \,dx < \underline{\int}_a^b f(x) \, dx,$$

Since the lower integral on the RHS is a supremum over partitions, there exists a partition $P$ such that

$$\underline{\int}_a^b [f(x)+g(x)] \, dx - \underline{\int}_a^b g(x) \,dx < L(P,f) \leqslant \underline{\int}_a^b f(x) \, dx.$$

Rearranging again,

$$\underline{\int}_a^b [f(x)+g(x)] \, dx - L(P,f) < \underline{\int}_a^b g(x) \, dx.$$

Reasoning as before, there exists a partition $P’$ such that

$$\underline{\int}_a^b [f(x)+g(x)] \, dx - L(P,f) < L(P’,g) \leqslant \underline{\int}_a^b g(x) \, dx,$$ and $$\underline{\int}_a^b [f(x)+g(x)] \, dx < L(P,f) + L(P’,g) .$$

Take a common refinement of the partitions $Q = P \cup P'$. Lower sums increase as partitions are refined and we have $L(Q,f) \geqslant L(P,f)$ and $L(Q,g) \geqslant L(P’,g).$

It follows that

$$L(Q,f+g) \leqslant \underline{\int}_a^b [f(x)+g(x)] \, dx < L(P,f) + L(P’,g) \leqslant L(Q,f) + L(Q,g).$$

This contradicts inequality (1) for lower sums, and, therefore

$$\underline{\int}_a^b f(x) \, dx + \underline{\int}_a^b g(x) \, dx \leqslant \underline{\int}_a^b [f(x) + g(x)] \, dx. $$

Second Lemma

Let $P$ be an arbitrary partition of $[a,b].$ Now $P$ may or may not include the point $c$. If not, then consider a refined partition $P'$ that includes $c$. Let $P' = P_1 \cup P_2$ where $P_1$ and $P_2$ are partitions of $[a,c]$ and $[c,b]$, respectively.

Then $$\overline{\int}_a^c f(x) \, dx + \overline{\int}_c^b f(x) \, dx \leqslant U(P_1,f) + U(P_2,f) = U(P',f) \leqslant U(P,f).$$

Now take the infumum over all partitions $P$ to obtain

$$\overline{\int}_a^c f(x) \, dx + \overline{\int}_c^b f(x) \, dx \leqslant \inf_{P} \,\,U(P,f) = \overline{\int}_a^b f(x) \, dx .$$

Answer 2

Let $\epsilon >0$ and note that there are partitions $P, Q$ such that

$\tag1 \underline{\int_{a}^{b}}{f(x)dx}-\epsilon<L(P,f)$

$\tag2\underline{\int_{a}^{b}}{g(x)dx}-\epsilon <L(g,Q)$

These inequalities persist if we pass to a common refinement, $T$, so

$\tag 3\underline{\int_{a}^{b}}{f(x)dx}+\underline{\int_{a}^{b}}{g(x)dx}-2\epsilon<L(T,f)+L(T,g)$.

But also, \begin{equation*}L(T,f) + L(T,g)\leq L(T,f+g)\leq\underline{\int_{a}^{b}}{(f+g)(x)dx} \tag{4}\end{equation*}

from which we get

$\tag5\underline{\int_{a}^{b}}{f(x)dx}+\underline{\int_{a}^{b}}{g(x)dx}-2\epsilon<\underline{\int_{a}^{b}}{(f+g)(x)dx}$

which is what we want.

For the second part, here is a hint: observe that if $P$ is any partition of [a,b], we may assume without loss of generality, that $c\in P$. Then $P'=P\cap[a,c]$ and $P''=P\cap[c,b]$ are parititions of $[a,c]$ and $[c,b]$,resp. and $P=P\cup P''$.