Mizar Commentary on Walter Rudin's Principles of Mathematical Analysis/The Riemann-Stieltjes Integral

In Mizar the Riemann-Stieltjes integral wasn't formalized until recently, so this chapter will primarily deal with that variant. Everything about the Riemann-Stieltjes integral in Mizar at the time of writing is in INTEGR22 and INTEGR23.

Definition and Existence of the Integral
6.1 Definition A partition is described by a Division (INTEGRA1:def 2), leaving $$x_0$$ out and changing $$\leq$$ to $$<$$. The intervals can be accessed via divset(P,i) (INTEGRA1:def 4), therefore $$\Delta x_i$$ is given by vol divset(P,i) (INTEGRA1:def 5). $$U(P,f)$$ is <tt>upper_sum(f,P)</tt> (<tt>INTEGRA1:def 8</tt>), $$L(P,f)$$ is <tt>lower_sum(f,P)</tt> (<tt>INTEGRA1:def 9</tt>). $$\textstyle\overline{\int}^b_a f \,dx$$ is <tt>upper_integral(f)</tt> (<tt>INTEGRA1:def 14</tt>), $$\textstyle\underline{\int}^b_a f \,dx$$ is <tt>lower_integral(f)</tt> (<tt>INTEGRA1:def 15</tt>). $$f$$ being <tt>integrable</tt> is in <tt>INTEGRA1:def 16</tt> (see also <tt>INTEGRA5:def 1</tt>), $$\textstyle{\int}^b_a f \,dx$$ is <tt>integral(f)</tt> (<tt>INTEGRA1:def 17</tt>, see also <tt>INTEGRA5:def 4</tt> for explicit integration bounds and <tt>INTEGRA7:def 2</tt> for indefinite integral). The inequalities are given by <tt>INTEGRA1:25,28,27</tt>.

6.2 Definition in <tt>INTEGR22</tt>.

6.3 Definition Refinement is <tt>INTEGRA1:def 18</tt>. No reference for definition of common refinement, but it is given by <tt>INTEGRA1:47</tt> and more precise by <tt>INTEGRA3:21</tt>.

6.4 Theorem is <tt>INTEGRA1:41/40</tt>.

6.5 Theorem is <tt>INTEGRA1:49</tt>.

6.6 Theorem is implicitly given by <tt>INTEGRA4:12</tt>.

6.7 Theorem No reference.

6.8 Theorem is <tt>INTEGRA5:11</tt>, for Riemann-Stieltjes <tt>INTEGR23:21</tt>.

6.9 Theorem is <tt>INTEGRA5:16</tt>.

6.10 Theorem No reference.

6.11 Theorem No reference. #TODO Reference that if $$f$$ is bounded and integrable and $$\phi$$ is continuous, then $$\phi\circ f$$ is integrable.

Properties of the Integral
6.12 Theorem (a) is <tt>INTEGRA1:57</tt> or <tt>INTEGRA6:11</tt> and <tt>INTEGRA2:31</tt> or <tt>INTEGRA6:9</tt>. (b) is <tt>INTEGRA2:34</tt>. (c) is <tt>INTEGRA6:17</tt>. No reference for something like $$\textstyle\int_{A\cup B}f\,dx=\int_A f\,dx + \int_B f\,dx$$ with $$A\cap B=\emptyset$$. (d) No direct reference, but 6.13(b) improves the bound anyway. (d) No reference.

6.13 Theorem (a) is <tt>INTEGRA4:29</tt>. (b) is <tt>INTEGRA4:23</tt> or <tt>INTEGRA6:7/8</tt>.

6.14 Definition $$I$$ equals <tt>chi(REALPLUS,REAL)</tt> (<tt>FUNCT_3:def 3</tt>, <tt>MUSIC_S1:def 2</tt>).

6.15 Theorem No reference.

6.16 Theorem No reference.

6.17 Theorem No reference. #TODO Relation between the Riemann and the Riemann-Stieltjes integral.

6.18 Remark No reference.

6.19 Theorem (change of variable) No reference. #TODO Find reference!

Integration and Differentiation
6.20 Theorem <tt>INTEGRA6:28/29</tt>, but no reference for the continuity of $$F$$ if $$f$$ is only integrable.

6.21 The fundamental theorem of calculus is <tt>INTEGRA7:10</tt>.

6.22 Theorem (integration by parts) is <tt>INTEGRA5:21</tt> or <tt>INTEGRA7:21/22</tt>.

Integration of vector-valued Functions
6.23 Definition Given by <tt>INTEGR15:def 13/14</tt> and <tt>INTEGR15:def 16-18</tt>. Theorem 6.12 (a) translates to <tt>INTEGR15:17/18</tt>. Theorem 6.12 (c) translates to <tt>INTEGR19:8</tt>. No reference for a translation of Theorem 6.12 (e) or Theorem 6.17, since we're still looking at Riemann integrals only.

6.24 Theorem No direct reference, but with continuity given by <tt>INTEGR19:55/56</tt>.

6.25 Theorem is <tt>INTEGR19:20/21</tt>.

Rectifiable Curves
6.26 Definition More generally, $$\gamma$$ being a <tt>parametrized-curve</tt> is defined in <tt>TOPALG_6:def 4</tt>, no reference for closed curves there. For simple closed curves in $$R^2$$ see the attribute <tt>being_simple_closed_curve</tt> (<tt>TOPREAL2:def 1</tt>). In pursuing the proof of the Jordan Curve Theorem a lot of articles have been written dealing with that attribute, in contrast to <tt>parametrized-curve</tt>. Also see <tt>INTEGR1C:def 3</tt> for a formalization of $$\mathcal{C}^1$$-curves using another approach. No reference for $$\Lambda(\gamma)$$ or rectifiable.

6.27 Theorem No reference.

Exercises
1. No reference. 2. No reference. 3. No reference. 4. No reference. 5. No reference. 6. No reference. 7. No reference. 8. See <tt>INTEGR10</tt> for the definitions. No reference for exercise. 9. No reference. 10. No reference. 11. No reference. 12. No reference. 13. No reference. 14. No reference. 15. No reference. 16. No reference. 17. No reference. 18. No reference. 19. No reference.