(d) Use (c) to show that if f and g are integrable then max{f
If in addition
g(x)dx. Theorem 2.2. Let f and g be integrable functions on [a b]. Then fg is an integrable function on [a
20-Mar-2019 (b)Prove that if f is integrable on [a b]
5 Let f and g be integrable functions on [a b]. (a) Show that if P is any partition of [a
The preceding property asserts that µ is an additive set function: µ([a b]) = µ([a
Properties of Integrable Functions. Theorem 5.7 (Linear Property). If fg are integrable on [a
then f = g almost everywhere and g is Riemann Integrable. = Lebesgue ... If f is Riemann-integrable then f is Lebesgue- integrable and. ∫. [a
(b) If fg are integrable then so is fg and f g.
(d) Use (c) to show that if f and g are integrable then max{f
If in addition
A bounded function f on [a b] is said to be (Riemann) integrable if L(f) g(x)dx. Theorem 2.2. Let f and g be integrable functions on [a
7(b) Show that if f is integrable on [a b]
Integrability is a less restrictive condition on a function than example if f
Function additivity. If fg are Riemann-integrable on [a
5 Let f and g be integrable functions on [a b]. (a) Show that if P is any partition of [a
LEMMA: f is integrable iff for every ? there exists a partition P such that If f g are integrable then also the product fg
D. 7.5. Simple consequences. Theorem 7.10. (a) If f and g are integrable on [a b] then f +g is
From the definition f is integrable if and only if
If f;g: [a;b] !R and both f and gare Riemann integrable then fgis Riemann integrable Proof Apply the Composition theorem The function h(x) = x2 is continuous on any nite interval Then h f= h(f) = f2 and h g= h(g) = g2 are Riemann integrable Also (f+g)2 is Riemann integrable (why?) Therefore fg= 1 2 [(f+ g)2 f2 g2] is Riemann integrable
INTEGRABLE FUNCTIONS Then g is a monotonic function from [ab] to R?0 Hence by theorem 7 6 g is integrable on [ab] and Z b a g = Ab a(g) Now let {Pn} be a sequence of partitions of [ab] such that {µ(Pn)} ? 0 and let {Sn} be a sequence such that for each n in Z+ S n is a sample for Pn Then {X (gPnSn)} ? Ab a(g) (8 5) If Pn
ProofofCorollary2: The integrability of f and fn both follow directly from Theorem 1 with ?(y) = y and ?(y) = yn respectively If f and g are bounded and integrable then so is f +g Hence by Theorem 1 with ?(y) = y2 we have that f 2 g2 and (f +g) = f2+g 2+2fg are all integrable The integrability of fg = 1 2 (f+g)2?f ?g2 now
(f +g) = ? b a f +? b a g: Proof f +g is integrable on [a; b]; because for any partition P U(f +g;P)?L(f +g;P) ? U(f;P)?L(f;P)+U(g;P)?L(g;P); and the right side can be made arbitrarily small To prove additivity choose P so that U(f;P)? " 2 < ? b a f < L(f;P)+ " 2 and U(g;P)? " 2 < ? b a g < L(g;P)+ " 2: Then ? b a (f +g
To prove that fg is integrable when f and g are simply note that fg= (1=2) (f +g)2f2g2!: Property (6) and the linearity of the integral then imply fg is integrable (8) If there exists m;M such that 00 such that m jf(x)j M for all x2[a;b] Note that 1 f(x) 1 f(y) = f(y)f(x) f
Theorem 1 If f and g are integrable on [a;b] then the product fg is integrable on [a;b] Proof First notice that fg = 1 2 (f +g)2 f2 g2: So it is enough to prove that if f is integrable then f2 is integrable If f is integrable then jfj is integrable Also f2 = jfj2 Let M = supjfj over [a;b] Suppose that P is a partition of [a;b] and I