Gauge Institute Journal, Volume 7, No 1, February 2011 H. Vic Dannon Lebesgue Integration H. Vic Dannon [email protected] February, 2010 Abstract: We show that 1. the Riemann integral may exist when the Lebesgue integral does not 2. the Fundamental Theorem of Calculus may hold for Riemannian Integration but not for Lebesgue integration of the same function 3. The same limit function of Integrable functions may be Riemann-Integrable but not Lebesgue integrable. Therefore, contrary to Common belief, Riemann Integration is more general than Lebesgue Integration. Furthermore, we show that a convergent sequence of measurable functions may have a non-measurable limit function. This cast a doubt on the validity of the main results of Lebesgue theory. Consequently, Riemannian Integration is not only a superior Integration Theory. It is the only Integration Theory that we have. 1 Gauge Institute Journal, Volume 7, No 1, February 2011 H. Vic Dannon Keywords: Lebesgue, Riemann, Integration, Measureable, Measure, Integrability, Completion, Banach spaces, Lp spaces, Dominant Convergence, Monotone Bounded Convergence, Fatou Lemma, Fubini, Tonelli, Beppo-Levi, Dirichlet function, Riemann Function, 2000 mathematics subject Classification: 28A20, 28A25, 28A35, 26A15, 26A30, 26A36, 26A42, 26A46, 26A48, Contents Introduction 1. Riemann Integral may exist when Lebesgue’s does not 2. The Fundamental Theorem of Calculus may hold for Riemann Integration but not for Lebesgue Integration of the same function 3. The same limit function of Integrable Functions may be Riemann-Integrable but not Lebesgue-Integrable 4. A Countable Set need not have Measure Zero 5. The Rationals in [0,1] are Non-Measurable 6. Dirichlet Function 7. Riemann’s Function 2 Gauge Institute Journal, Volume 7, No 1, February 2011 H. Vic Dannon 8. A convergent sequence of Lebesgue Measurable, Integrable Functions may have Non-measurable, Non-integrable Limit Function 9. L1 is an Incomplete Normed Linear Space 10. The Completion of L1 to a Banach Space has Lebesgue- Non-Integrable Functions 11. Dominant Convergence 12. Lebesgue Monotone Bounded Convergence 13. Fatou’s Lemma 14. Term by Term Series Integration 15. Iterated Lebesgue Integration 16. Riemann Integral generalizes Lebesgue’s 17. The Space of Riemann Integrable Functions 18. Cantor’s Function 19. The Meaning of a Measurable Set. 20. The meaning of a Measurable Function References 3 Gauge Institute Journal, Volume 7, No 1, February 2011 H. Vic Dannon Introduction. A continuous function is Cauchy integrable, if and only if the Sums that approximate the area under the function graph f(t )(x −x )+ f(t )(x −x )+....+ f(t )(x −x ), 1 1 0 2 2 1 n n n−1 converge to a limit, as the size of the subintervals of the partition a = x < x < ... < x = b approaches zero. 0 1 n In the Cauchy Sums, t ,t ,...,t are the endpoints of the partition 1 2 n subintervals. Riemann allowed a bounded function that may have infinitely many discontinuities, and let t ,t ,...,t be arbitrarily chosen 1 2 n points in each subinterval. Riemann required that the oscillation sums max f(t)− min f(s) (x −x )+....+ max f(t)− min f(s) (x −x ) 1 0 n n−1 x ≤t≤x x ≤s≤x x ≤t≤x x ≤s≤x 0 1 0 1 n−1 n n−1 n will vanish as the size of the subintervals of the partition a = x < x < ... < x = b approaches zero. 0 1 n Lebesgue allowed a measurable function, and sequences of discontinuities. But restricted his integral by requiring the measurable function to be absolutely integrable. We shall recall that similarly to series convergence, there are functions that are Riemann Integrable, but not absolutely integrable. 4 Gauge Institute Journal, Volume 7, No 1, February 2011 H. Vic Dannon Lebesgue theory is based on the belief that any countable set has measure zero, so that the integral on a countable set is zero. We recall that sequencing and measurability are unrelated. Some countable sets have non-zero measure, and some countable sets have no measure at all. Consequently, Riemann presented two Functions that satisfy Riemann’s Oscillation sums condition, and are Riemann Integrable, but not Lebesgue integrable. Furthermore, the Dirichlet function, that is not Riemann Integrable, is not Lebesgue integrable either. This suggests that Riemann’s integral generalizes Lebesgue’s. While Riemann-integrability is preserved for the limit function under uniform convergence, Lebesgue theory claims that measurability is always preserved for the limit function. We disprove Lebesgue’s theory claim, with a convergent sequence of measurable functions, that does not converge to a measurable function. That is, we show that the Dirichlet Function is a Non- Measurable limit of measurable functions, and has no Lebesgue Integral. On the other hand, Riemann’s Function is Riemann integrable over its Non-Measurable set of discontinuities. 5 Gauge Institute Journal, Volume 7, No 1, February 2011 H. Vic Dannon These facts disprove the main results of Lebesgue Theory: In particular, 1. L1 has Cauchy sequences of Lebesgue Measurable Integrable functions with Lebesgue Non-Measurable, Non-Integrable Limits. Hence L1 is an Incomplete, Normed Linear space, and its Completion to a Banach Space has Non-Measurable, Non-Integrable functions 2. the Lebesgue function spaces Lp, 1 ≤ p ≤ ∞, have Cauchy sequences of Riemann integrable functions with a limit function that is not measurable, and not Lebesgue integrable, and the Lp spaces are incomplete. 3. Lebesgue Dominant Convergence Theorem is invalid. Only the Bounded Convergence of Arzela for Riemann- Integrable functions holds. 4. Lebesgue Monotone Bounded Convergence is invalid. 5. Fatou’s Lemma is invalid 6. Beppo-Levi term by term integration of a series of Lebesgue integrable functions- is invalid 7. Fubini and Tonelli Theorems for iterated Lebesgue integration are invalid. Consequently, Riemannian Integration is not just a superior Theory. It is the only Integration Theory that we have. 6 Gauge Institute Journal, Volume 7, No 1, February 2011 H. Vic Dannon 1. Riemann Integral may exist when Lebesgue’s does not Many series that converge conditionally, do not converge absolutely. The alternating series 1− 1 + 1 − 1 +... 2 3 4 converges to log2, and the alternating series 1− 1 + 1 − 1 +... 3 5 7 converges to 1π. 4 But the absolute values series diverge. Similarly, the requirement of absolute integrability imposed on the Lebesgue integral, eliminates functions that are Riemann Integrable. sinx For instance , over the interval 0 ≤ x < ∞, is Riemann x Integrable, but not absolutely integrable. Therefore, similarly to the series, 1.1 Riemann’s conditional integrability allows integrability that is not allowed by Lebesgue’s absolute integrability. 7 Gauge Institute Journal, Volume 7, No 1, February 2011 H. Vic Dannon We establish this by either 1.2, or 1.3. sinx 1.2 is Riemann integrable but not Lebesgue integrable over x [0,∞) x=∞ sinx π Proof: It is well known that ∫ dx = . x 4 x=0 sinx Therefore, is Riemann Integrable over [0,∞).(cid:44) x sinx But is not absolutely integrable over[0,∞). Indeed, x x=∞ x=π x=2π x=3π x=4π sinx sinx sinx sinx sinx ∫ dx = ∫ dx + ∫ dx + ∫ dx + ∫ dx +... x x x x x x=0 x=0 x=π x=2π x=3π x=π x=2π x=3π x=4π 1 1 1 1 ≥ ∫ sinxdx + ∫ sinxdx + ∫ sinxdx + ∫ sinxdx +... π 2π 3π 4π x=0 x=π x=2π x=3π x=π x=2π x=3π x=4π 1 1 1 1 = ∫ sinxdx + ∫ −sinxdx + ∫ sinxdx + ∫ −sinxdx +... π 2π 3π 4π x=0 x=π x=2π x=3π 1 0 1 2π 1 2π 1 4π = cosx + cosx + cosx + cosx +... π (cid:8)(cid:11)(cid:11)(cid:9)(cid:11)(cid:11)π(cid:10) 2π (cid:8)(cid:11)(cid:11)(cid:9)(cid:11)π(cid:11)(cid:10) 3π (cid:8)(cid:11)(cid:11)(cid:9)(cid:11)3(cid:11)π(cid:10) 4π (cid:8)(cid:11)(cid:11)(cid:9)(cid:11)3(cid:11)π(cid:10) 2 2 2 2 2⎛ 1 1 1 ⎞ = π⎜⎜⎜⎝1+ 2 + 3 + 4 +...⎠⎟⎟⎟⎟ = ∞.(cid:44) 1 1 1.3 sin is Riemann integrable, but not Lebesgue integrable, t t over (0,1]. 8 Gauge Institute Journal, Volume 7, No 1, February 2011 H. Vic Dannon Proof: x=1 x=1 1 1 1 1 ∫ sin dt = lim ∫ sin dt t t n→∞ t t x=0 x=1 n 1 The transformation x = yields t x=n sinx = lim ∫ dx n→∞ x x=1 x=∞ sinx = ∫ dx x x=1 x=∞ x=1 sinx sinx = ∫ dx − ∫ dx x x x=0 x=0 x=∞ x=1 sinx sinx Since both Riemann Integrals ∫ dx , and ∫ dx exist, x x x=0 x=0 x=1 1 1 1 1 ∫ sin dt exists, and sin is Riemann Integrable over (0,1].(cid:44) t t t t x=0 1 1 But sin is not absolutely integrable over (0,1]. We have t t x=1 x=1 1 1 1 1 ∫ sin dt = lim ∫ sin dt t t n→∞ t t x=0 x=1 n 1 The transformation x = yields t x=n sinx = lim ∫ dx n→∞ x x=1 9 Gauge Institute Journal, Volume 7, No 1, February 2011 H. Vic Dannon Similarly to the proof of 1.2, 2⎛1 1 1 ⎞ ≥ π⎜⎜⎜⎝2 + 3 + 4 +...⎟⎠⎟⎟⎟ = ∞. 1 1 Therefore, sin is not Lebesgue Integrable over (0,1].(cid:44) t t Therefore, 1.4 Riemann Integral may exist when Lebesgue’s does not Proof: 1.2, or 1.3.(cid:44) 10