Real Analysis/Darboux Integral

Another popular definition of "integration" was provided by Jean Gaston Darboux and is often used in more advanced texts, such as this wikibook, due to its introductory ease. In this chapter, we will define the Darboux integral, and demonstrate the equivalence of Darboux integrals and the more widely known Riemann integrals.

Construction
Unlike Riemann Integration, this version of the integral will forgo one assumption of the function &fnof; — that it must be continuous. It will only assume that the function &fnof; is bounded on [a,b]. Of course, normal assumptions for a Real Analysis course such as the function only operating on real numbers over the interval of focus can be presumed (i.e. $$f:[a,b]\to\mathbb{R}$$)

Partitions
We will modify the definition of partition for the Darboux Integral so that the values a and b are also included in the set. For completeness, we will write out this new definition again.

For now, we will ignore the actual process of indexing these values. However, it should be noted that our definition of the partition does not make a claim about the relationship between the numbers; these values are not necessarily evenly distributed - but they can.

Upper and Lower Sums
Let $$\mathcal{P}$$ be a partition of $$[a,b]$$

For every $$x_i\in\mathcal{P}$$, you can define two special numbers:


 * $$m_{i} \, \dot{=} \, \inf{\{f(x) \,|\, x\in [x_{i-1}, x_i]\}}$$ and


 * $$M_{i} \, \dot{=} \, \sup{\{f(x) \,|\, x\in [x_{i-1}, x_i]\}}$$

The verbal definition of these two variables is more clear; mi defines the infimum of the set of valid &fnof;(x) values in between two partition points and Mi defines the supremum of the set of valid &fnof;(x) values in between two partition points.

Next, we will define the key functional component of the Darboux Integral, the sums.

Borrowing from geometry, you will notice that both sums are essentially additions of various rectangular shapes that are tied to the function &fnof; due to the definition of length being either a supremum or infimum respectively.

It is important to note that although the upper and lower sum borrows function notation, it is not, necessarily, a function in the normal sense. It takes partitions as the input, which size is a natural number. The function &fnof; is treated as a fixed constant.

There is actually only one more construction required in order to reach the Darboux integral. The only problem? This last step is to relate the upper sum and the lower sum. After all, the rectangles generated from this function leave out a lot of gaps because there are too few partition points. The more partition points there are, the more that Mi and mi converge upon the same value. The next task should be clear now; we need to prove that the upper sum and lower sum can converge onto a point.

Refinement
The second last piece of the construction requires that we prove the following two lemmas regarding our partition and our sums:


 * 1) $$L(f,\mathcal{P}) \le L(f,\mathcal{P}^*)$$
 * 2) $$U(f,\mathcal{P}) \ge U(f,\mathcal{P}^*)$$

Excuse me, we have to define what the partition P with the asterisk means first before we can analyze this statement.

Okay, why do we need to prove this? Simple, these inequalities state that more partitions leads to a better approximation of the actual area. The lower bound will increase as it reaches the "area", while the upper bound will decrease as it reaches the "area". This should be a fact so intuitive that the idea of proving it might have never crossed your mind. However, we will prove this lemma right now. It will be needed for the final piece of the Darboux Integral puzzle.

This proof is simple and will only require inequality algebra.

Similarly, we can prove $$U(f,\mathcal{P})>U(f,\mathcal{P}^*)$$ using the same method, by inverting the necessary functions.

Convergence
Now that we have proved that our intuition is correct; more partitions will yield an even closer approximation from both the lower sum and the upper sum, it is only fair to see if we can bring them together. If I can use mathematical symbols freely, it can be depicted as


 * $$L(f,\mathcal{P}_1) \rightarrow \text{``Area''} \leftarrow U(f,\mathcal{P}_2)$$

when the upper sum (the area of overestimation) is larger than the actual "area" and the lower sum (the area of underestimation) is smaller, yet both converge upon the "area" when the partition becomes finer. However, thinking like this will lead us to avoid the mathematical pieces we have collected that can also as fairly construct our integral. The roadmap to prove the Darboux Integral leads us to the final piece,


 * $$L(f,\mathcal{P}) \le U(f,\mathcal{P})$$

where $$\mathcal{P}$$ can be thought of as a partition full enough to yield the perfect approximation. We will call it, for this explanation portion, a perfect partition, although the perfect partition is importantly not infinite. However, you might be wondering how to solve this; the previous lemma does not make any comparisons between the lower and upper sum. That is why we are going to prove this instead:


 * $$L(f,\mathcal{P}_1) \le U(f,\mathcal{P}_2)$$

when $$\mathcal{P}_1$$ and $$\mathcal{P}_2$$ are any partitions of [a,b]. Yes, they do not need to be the same partition, as long as they are over the same interval [a,b]. This is actually going to be simpler, because our proof will use these two sums like bounds — dare I compare it as a squeeze?

Conclusion
We come at an impasse. Our final piece yields a very strange answer about the lower and upper sum. Namely that they are not an equality, but an inequality
 * $$L(f,\mathcal{P}) \le U(f,\mathcal{P})$$

where the certainty of the number remains unknown. However, we can easily sidestep this issue by breaking it into two cases and validating one or the other. What do we mean by validation? We can define the integral, namely the Darboux Integral, as being the number ensuring the equality of the upper and lower sum. We can then define an invalid integral as maintaining the inequality. In mathematical notation, we define the integral as being
 * $$L(f,\mathcal{P}) = U(f,\mathcal{P})$$

and rejecting every other case as being an invalid integral.

From here, we completed the construction of the Darboux Integral from the bottom-up.

Remarks

 * 1) Of course, the function has to be real i.e. $$f:[a,b]\to\mathbb{R}$$.
 * 2) The Darboux Integral is defined on the condition of uniqueness, unlike other concepts in this wikibook, such as limits, that are implied from the definition.

Properties
Let $$f:[a,b]\to\mathbb{R}$$

$$f$$ is Darboux integrable over $$[a,b]$$ if and only if for every $$\varepsilon>0$$, there exists a partition $$\mathcal{P}$$ on $$[a,b]$$ such that $$U(f,\mathcal{P})-L(f,\mathcal{P})<\varepsilon$$

Proof
($$\Rightarrow$$)Let $$A=\int_a^b f$$ and let $$\varepsilon>0$$ be given. Thus, by Gap Lemma, there exists a partition $$\mathcal{P}$$ such that both $$U(f,\mathcal{P}),L(f,\mathcal{P})\in V_{\frac{\varepsilon}{2}}(A)$$, and hence $$U(f,\mathcal{P})-L(f,\mathcal{P})<\varepsilon$$

($$\Leftarrow$$)Let $$\mathcal{P}_0$$ be any partition on $$[a,b]$$. Observe that $$L(f,\mathcal{P}_0)$$ is a lower bound of the set $$\mathcal{U}=\{U(f,\mathcal{P})|\mathcal{P}$$ is any partition$$\}$$ and that $$U(f,\mathcal{P}_0)$$ is an upper bound of the set $$\mathcal{L}=\{L(f,\mathcal{P})|\mathcal{P}$$ is any partition$$\}$$

Thus, let $$\alpha=\sup\mathcal{L}$$ and $$\beta=\inf\mathcal{U}$$. As $$L(f,\mathcal{P})\beta$$ cannot be true. Also, as $$\alpha,\beta$$ are a supremum and infimum respectively, $$\alpha<\beta$$ is also not possible. Hence, $$\alpha=\beta=L$$ (say).

As $$L=\sup_{\mathcal{P}} L(f,\mathcal{P})=\inf_{\mathcal{P}} U(f,\mathcal{P})$$, we have that $$\int_a^b f=L$$

Equivalence of Riemann and Darboux Integrals
At first sight, it may appear that the Darboux integral is a special case of the Riemann integral. However, this is illusionary, and indeed the two are equivalent.

Lemma
(1) Let $$f:[a,b]\rightarrow\mathbb{R}$$ be Darboux Integrable, with integral $$L$$ Define function $$\varepsilon (\delta)=\sup\{|L-S(f,\dot{P})|:\|\dot{P}\|=\delta\}$$

(2) Then $$\delta_1<\delta_2$$ $$\Rightarrow$$ $$\varepsilon(\delta_1)<\varepsilon(\delta_2)$$

Proof
Let $$\delta_1<\delta_2$$. Consider set $$T$$ of tagged partitions $$\dot{P}$$ such that $$\varepsilon(\delta_1)\leq|L-S(f,\dot{P})|$$

Let $$T'$$ be the set of $$\dot{P'}$$ where $$\dot{P'}\subset\dot{P}\in T$$ and $$\|\dot{P'}\|=\delta_2>\delta_1$$

note that $$T'\neq T$$ and that the set $$T'$$ indeed contains all partitions $$\dot{P'}$$ with $$\|\dot{P'}\|=\delta_2$$

Now, for $$\dot{P}\in T$$, we can construct $$\dot{P'}\in T'$$ such that $$|L-S(f,\dot{P'})|>|L-S(f,\dot{P})|$$

Hence, $$\displaystyle\sup_{\dot{P}\in T}\{|L-S(f,\dot{P})|\}<\sup_{\dot{P'}\in T'}\{|L-S(f,\dot{P'})|\}$$

i.e. $$\varepsilon(\delta_2)>\varepsilon(\delta_1)$$

Theorem
Let $$f:[a,b]\rightarrow\mathbb{R}$$

(1)$$f$$ is Riemann integrable on $$[a,b]$$ iff

(2)$$f$$ is Darboux Integrable on $$[a,b]$$

Proof
($$\Rightarrow$$) Let $$\epsilon>0$$ be given.

(1)$$\Rightarrow$$ $$\exists$$ tagged partition $$\dot{P}$$ such that $$|S(f,\dot{P})-L|<\frac{\epsilon}{2}$$.

Let partitions $$P_1$$ and $$P_2$$ be the same refinement of $$\dot{P}$$ but with different tags.

Therefore, $$|S(f,P_1)-S(f,P_2)|<\epsilon$$ $$\forall$$ $$P_1$$ and $$P_2$$

i.e., by the triangle inequality, $$|S(f,P_1)|-|S(f,P_2)|<\epsilon$$

Gap Lemma $$\Rightarrow$$ $$U(f,P)-L(f,P)<\epsilon$$,

$$\epsilon>0$$ being arbitrary, using Theorem 2.1, we have that $$f$$ is Darboux Integrable.

($$\Leftarrow$$)Let $$\epsilon>0$$ be given.

(2), Theorem 2.1 $$\Rightarrow$$ $$\exists$$ partition $$P$$ such that $$U(f,P)-L(f,P)<\epsilon$$

Hence, $$|L-S(f,P)|<\epsilon$$ as $$L(f,P)\leq S(f,P)\leq U(f,P)$$

By Lemma 3.1, $$|L-S(f,P')|<\epsilon$$ if $$\|P'\|<\|P\|$$

Thus, if we put $$\delta=\|P\|$$, we have (1)

We note here that the crucial element in this proof is Lemma 3.1, as it essentially is giving an order relation between $$\varepsilon$$ and $$\delta$$, which is not directly present in either the Riemann or Darboux definition.