Question

Let f : [0, 1] x [0, 1] + R be defined by f(x, y) = {1 if y = 23, 0 if y + x2 Show that f is integrable on (0, 1] x [0, 1]. You may take the previous problem as given

Answer 1

bounded f . function of [0, 1] x [0, 1] dominated Note that f is on [0] [91, f> and is dominated by the characteristic function of [0,1] * [91] since lebesgue lebesgue measure [91] *[0,1] is finite, the Characteristic function is integrable. Hence by lebesgue convergence theorem, f is integrable in the Lebesgue sense. Note : any measurable S., the characteristic functions of $, xg is defined by Xg(x) = {i s, if x 85 and. Sa, (x) = x(s), the I, if XES lebesgue measure of s. set For

we integrability we be area of the shaded region A2 Now must Check the Riemann of f. ie. for any EXO should find partion PE of [0,1] ×[o,i] such that UCPE, f) - LCPC, f) <6. So let Exo arbitrary. Az Let us form the nEIN. Ket a tricky way. yan we Consised the Curves y=xit ħ and y=x-h. cito) shall find the partition in AI Darth Az approximate supper bound ALE (C)、 Ś (a+ ta) de taa) 1 Fm 3 3 ) 11 N). tor 1/3 + 2 (5-) + 3nin - 2 - 2 25 so that Now choose 2 2 30 in 3 (Mx-mx). Ak where mk large enough 룩 <E Az <E. we claim that P {A,, that P { A., A₂, Az} is A3} is the required Partition, because UCPE,F) - LCPE, f) = Mk= Sup f(x,y) inf f(x,y) (1) E AK (1) E AK M, = m, =o=M3=m3. M₂ = 1, m₂ = 0 (by definition of f) U (PE, f) - LCP, f - Az <E (as required) So, fis Riemann integrable. (Note: A partition need not be the mesh or grid partition, it is Just the union of disjoint regular sets. Note that whereas So, always of type