A new characterization of /с-fold Lebesgue integralAbstract. A Perron type definition of the /с-fold Lebesgue integral is obtained utilizing the

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXVIII (1988) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRACE MATEMATYCZNE XXVIII (1988)

U. D

and A. G. D

(Kalyani, West Bengal)

A new characterization of /с-fold Lebesgue integral

Abstract. A Perron type definition of the /с-fold Lebesgue integral is obtained utilizing the kth absolute continuity concept of Das and Lahiri [5]. It is shown that such an integral is an ACk function and that every ACk function is the integral of its own kth order derivative and also that the integral is equivalent to the fc-fold Lebesgue integral.

1. Introduction and preliminaries. Bullen [2], De Sarkar and Das [9], De Sarkar, Das and Lahiri [10] and some others obtained definitions of higher order Perron, Denjoy and Ridder type integrals and their approxi­

mate extensions. Bullen [2] showed that his /^-integral is the k-fold Cesàro- Perron integral of Burkill [3] starting with the classical Denjoy-Perron integral, and the integral is a certain generalized AC*, [zlC*G], function. In this paper we show, as a natural demand, that a к-fold Lebesgue integral is an ACk function and, in fact, an AC* function, /^-integral of Bullen [2] and that of De Sarkar and Das [9] are then Perron type generalizations of к-fold Lebesgue integral. Following Kenedy [13], we obtain a direct definition of kth order Lebesgue integral which we call the Lk-integral. Indeed, we may utilize ACk above and below functions of Das and Lahiri [5] and adapt Ridder’s method of defining major and minor functions so as to obtain the definition of upper and lower Lk-integrals. It is shown that an LMntegral is necessarily an ACk function and that every ACk function is the Lk-integral of its own kth derivative. Finally, it is shown that an Lk-integral is equivalent to the к-fold integral.

We require some known definitions and results.

The functions that occur are real single-valued functions defined in the closed interval [a, b] and к is a positive integer greater than 1. If x0, x i, ...» xk are any k + 1 distinct points, not necessarily in the linear order, in [я, b], then the kth divided difference of / is defined by

к к

Qkif'i x0> •••> xk) ~ X \ f (X ')/ П (^i ■*"./)} *

i= 0 j = 0

}*i

Let x0 be any fixed point in [a, b] and let , ..., xk be any set of к

distinct points in [a, b] with the property 0 < |xx — x0| < |x2 — x0| < ...

. . . < |xk —x0|. If the iterated limit

lim ... lim k \ Q k{f; x 0, x x, xk) xk -*xo xi *o

exists (possibly infinite), then this limit is called the kth Riemann* derivative of / at x0 and is denoted by Dkf ( x 0). Taking lim sup (respectively, lim inf) at the kth stage, we get the upper (respectively, lower) derivatives Dkf (x0) (respectively, Dkf ( x 0)). The one-sided derivatives Dk+ / ( x 0), DlL / ( x 0), etc., are obtained in the usual way by taking all the points x l5 x2, ..., xk on the same side of x0. We note that Dk+ f (x0) = D t / ( x 0) does not necessarily imply the existence of Dkf ( x 0). If, however, in addition Dk~ lf ( x 0) exists, the existence of Dkf (x0) is ensured.

The ordinary (resp. Peano) derivative of / of order к at x will be denoted b y / (M(x) (resp. /(k)(x)). By a set we shall mean a subset of [a, b] and the Lebesgue measure of a set A will be denoted by mA.

If for all choices of k+ 1 distinct points x0, x l5 ..., xfc in E we have Qk(f', x0, x x, ..., xk) ^ 0, then / i s called к-convex on E. If —/i s к -convex on E, then / is called k-concave on E.

The number

n — к

Vk( f ; E) = sup X (Xi+*-x(-)|6 * (/; xf, xi + 1, ..., x/+Jk)|, я (=0

where the supremum is taken for all n subdivisions in E of the form x0 < x x

< ... < x„, x ;e £ , i — 0, 1, ..., n, is called the total kth variation of / on E. If Vk{ f \ E ) < +oo, then / is said to be of bounded k-th variation, BVk, on E.

Let X j o ^ ^1,1 x l,k ^ -^2,0 ^2,1 ^ ^ x 2 ,k ^ ^ xn,o < xn>1

< . . . < x nk be any subdivision of E, xl7 e £ . The intervals (xi>0, xik), i = 1, 2, ..., n, are said to form an elementary system I, say, in E and is denoted by

/ ( x u , ..., xi i b l ): (xi(o, xi>fc), i = 1, 2, ..., n.

V i —

X

0

•••»

i= 1

n

< r \ I \ =

X

0, •••,

i — 1 n

m l =

X

o)- /= i

Write

Characterization o f к-fold Lebesgue integral 49

The function / is said to be absolutely k-th continuous, ACk, on E if for arbitrary £ > 0 there exists 3 (e) > 0 such that for any elementary system / in E with ml < 3 the relation a |/| < £ is satisfied.

The function / is said to be ACk above (respectively below) on E if for arbitrary £ > 0 there exists 3 (fi) > 0 such that for any elementary system / in E with ml < 3 the relation o\I\ < e (resp. ol > — s) is satisfied.

The following definition is borrowed from De Sarkar and Das [9].

For any two real numbers c, d with c < d, define

S *i(/; c, d) = sup|Qk- t ( f ; x u x 2, ..., x * ) - ô * - i( /; c, x u ..., xk- t)\, 5k2( /; c,d) = sup\Qk- i ( f ; x u x 2, ..., хк_ 1? d ) - Q k. l ( f; x u x 2, ..., x*)|, where suprema are taken for all points xh i = 1 , 2 , . . . , k, with c < x t < x 2

< .. . < xk ^ d and c ^ Xj < x 2 < ... < xk < d.

Take

Sk( f ; c, d) = max {Sk l ( f; c, d), Sk2( f ; c, d)}.

The function / is said to be AC* on E if to each £ > 0 there corresponds a 3 > 0 such that for every finite sequence of non-overlapping intervals {{ch dt)}, ch dte E, ^ Б к( / ; с ь d f < e whenever £ ( 4 - c , ) < 3.

i i

If/ is AC$ on E, th e n /is ACk on E and if £ is a closed interval an ACk function becomes an AC% function. It further follows from Corollary 2.3 of De Sarkar and Das [9] that if / is ACk on [a, b], then f (k) is summable (L) on [a, b].

It is known [5], [ 8] that an ACk function is BVk and also that if f e A C k [a, b], then f {k~r) is*^Cr on [a, b], 1 ^ r < k - 1 and that f (k) exists a.e. in [a, b]. Further, i f / i s ACk above or below on [a, b], having bounded {k — l)th divided differences, then / eBVk [a, b].

2. k-fold Lebesgue integral. Russell [16], [17] proved the following theorem:

X

I f f is к-convex (B Vk) on [я, b], then F(x) = j f ( t ) d t is (k-\-l)-convex ( Щ +1) on la, b].

We obtain an analog.

X

T heorem 2.1. I f f is ACk on [a, b] (k ^ 1), then F (x) = J/ (dt) dt is

a

ACk+1 on la, b].

Proof. Consider any elementary system I ( x iA, ..., xitk): {xi>0, xitk+1), i = 1, 2 , . . . , n in la, b] and let e > 0 be arbitrary. Since/ eA C k la, b], there exists > 0 such that

4 - Roczniki PTM - Prace Matematyczne XXVIII

( 1 ⁾ Z lôk- 1 { f , 4 2> • * • » 4 k + l) Qk — 1 i f ’ 4 l ’ • * * » 4k)l ^ £/2 i= 1

whenever £ (xtfk + j — xi>0) < ^1.

f= l Now ( 2 ) Z ( ^{4 * + 1}

i= 1

' -4o) lôk+ 1 4 o . 4 l > • • • » -4fc + l)

i = 1

= 1 i = 1

/= 1

\ X - X u

) - Q k ( F ; x itо , . . . , x,„

1 4 2 » 4 k +1 ^ ~ ~ Q k - 1

Ч 2 ^{. •• •} » 4 k + 1

4 o ) ) ; 4 i > •• • > 4k)|>

F (x )-F (x ,,0)

x - x l(0 » 4 1 ’ • • • » 4 k

Since / is continuous and к is finite, there exists <52 > 0 such that for each i, i = 1, 2, n, the ith term on the right of (2) differs from the ith term on the left (1) by a number less than e/2i + 1 whenever (xi>k+1 — xl>0) < ô 2. Choose ô = min(di, nô2). Then, from (2), we obtain

n

Z ( 4 k +1 Xit0) \ Q k + 1 ( F , •Xj,0> 4 l ’ •••» 4 k + l)l

i= 1

< z \ Q k - i ( f ^ i a , . . . , x i t k+i ) - Q k_ 1 ( f ; x i t l , . . . , x itk) \ + Z £/2i + 1

/= l i= 1

< e/2 + e/2 = e whenever £ (xi>k+1 — xi>0) < à.

i= l This completes the proof.

We immediately obtain the following theorem.

T heorem 2.2. A к-fold Lebesgue integral on [a, b] is ACk on [a, b].

De Sarkar and Das [8] proved that iff e A C k [£ ], then Dk~ lf e A C [F]

for some set E, where the existence of Dk~ xf was taken over the set E. If E — [я, b],. then Dk~ lf is necessarily continuous and bounded in [a, b].

Utilizing Oliver [15] or Verblunsky [18], we obtain that if f e A C k \_a, b];

then / <к~ 1}e ЛС[a, b]. Here we obtain the reverse assertion.

T heorem 2.3. I f / (k_1) exists and is AC on [a, b], tbcn / is ACk on

la, b].

Characterization o f к-fold Lebesgue integral 51

Proof. Following Milne-Thomson [14], p. 6, we obtain, for any set of ( r + 1) points x0, x l5 ..., xr in [a, b],

r l Q r i f ; x0, • ••, *,-) = / (r)(£r), 1 < k - l ,

where £r is a number lying between the minimum and the maximum of {xf}.

Consider an arbitrary elementary system

7(xu , ..., x ^ .!) : (xi>0, xi>k), i = 1, 2, ..., n,

in [а, b]. There are £u , f (>2 respectively in (xi(0, хг>к_!) and (xitl, xik), i = 1, 2, ..., n, such that

(3) (k - \ ) \ { x u - x it0)Qk{ f ; xi>0, xi(1, ..., x ,k) = / (k- 1)( ^ 2) - / (t- 1)(^>1).

Without any loss of generality Же assume that < £if2, i = 1, 2, n. The intervals (xl>0, £u ), (fu , &>2), (£i>2, xa ), i = 1, 2, ..., n, are pairwise disjoint.

Let s > 0 be arbitrary. Since / (fc-1) is AC on [a, b], there exists S > 0 such that

i !l/l‘ " " K u ) - / “ " ll(4 o )l+ l/ ‘ " 1,« i,2) - / ,k" 1’fâ,i)l + i= 1

+ 1 f ' “- “ (« u )-/* * " " (W l} < s ( f c - 1)!

Л

whenever £ (xi>k — xl>0) < b.

i = 1

Consequently from (3) we obtain

S x ,,0 ... XU)| = £ | / №- 1, (6,2) - / i ‘ - 1'K u )I/(*-1)! < «

* = 1 1=1

n

whenever £ (x/>k — xf>0) < <5, and thus the theorem is proved.

/=i

Das and Lahiri [5] obtained the following theorem.

T heorem 2.4. I f f is ACk on [a, b] and Dkf vanishes almost everywhere in La, b], then f is a polynomial o f degree (Ac — 1) at most.

We establish the following generalization.

T heorem 2.5. Let Dk~ 1f exist on [a, b]. I f f is ACk below, A£_ k, on La, b] and D f f ^ 0 almost everywhere in [a, b], then f is к-convex on [a, b].

Proof. Let E = lx: a < x < b, D_kf ( x ) ^ 0]. Then mE = b — a. For each x e E and arbitrary £ > 0, there are sequences jxj*0}, i = —k, ..., — 1, 1, ..., к such that x(2k < ... < x("\ < x < xV° < ... < x(k , x(kn) — х(^к -» 0 and

(4> k'-Q A f; x<">, x f f , ... x t t k) > - e /2 ( k + l)( b - a ) ,

n ~ 1, 2, . . . , where we write x = x(0n).

It is clear that the closed intervals like [х("\, x*"*] associated with each x e £ for which condition (4) is satisfied cover the set E in Vitali’s sense.

Hence to each set ô > 0 we can select from them a finite number of pairwise disjoint closed intervals dt = [ x ^ .j , xu ] with x ,_ 1<k < x , _ k < . . . < x (>_ 1

< Xt.o < xui < ••• < *i,*; t = 0, 1, n, where xi>0 = x„ x0, - k = a, x„,k = b such that

(5) m ( £ - £ 4) <<5.

i = 0

Since х ,_ 1>к< х |(_к for j = 0, l , . . . , w , it is clear that the intervals [ x , _ k, xitk], i = 0, 1, n are pairwise disjoint.

Let j be any positive integer such that 1 ^ j < к. Consider the ele­

mentary system

(fi) l,j+1’ Xi,j-k- l)' (xi— 1J» Xi,j — k~ l)> * 2, ..., W, where x ,,.* .! = x ,_ 1>Jk and x, _1>k + 1 = x , , _ k.

As j ranges from 1 to k, we obtain к numbers of elementary systems in (6) one for each j.

Since / is ACk below on [я, b], we have

I t Л

Y. QiJ == Y (Xi,j~k~ 1 - 1 yj) Qk ( f > xi~l,jf •••> x i,j-k~l) '> в/2к i = 1 « = 1

for j = 1, 2, . . . , к and so

(7) i i Qu > -e/2.

j= 1 i= 1 On the other hand, by using (4),

Qi,j (Xi,j + k xi,j) Q k if » xi,j> •*•> x i,j + k) ^ e(Xi,j + k Xi,j) 2 ( k + l ) ( b - a ) ’

i = 0, 1, ..., и and / = - k , ..., 0. This gives

I e,-j > -

j=-k

e(Xi,j + k Xi,j)

2 ( b - a ) i = 0, 1, ..., n.

Consequently,

x i e ij > - e / 2.

i=0 j= -k

(

)

Characterization o f k-fold Lebesgue integral

Combining (7) and (8) and utilizing Lemma 4 of Russell [16], it follows that Qk — 1 (/> ^и,1> • • •» -^n,k) Qk- 1 i f у ^0, — ky • • •> ^0,— l)

-i iftj+i j= 1 i = 1 i - 0 j - - k ⁱ <&>-«•

It, therefore, follows that Dfl~ 1f ( b ) —Dk~ 1f ( a ) > —e. As e > 0 is arbitrary, we have Dk~ 1 f (b) ^ Dk~ 1 f (a). Applying the above argument to any sub­

interval [ х г, x2] of [a, 6], we obtain Dk~1f { x 2) ^ Z ^ ^ /fr i). Hence Dk~ if is non-decreasing on [a, b]. Since Dk~ 1f is the Peano derivative/k_ t) (Denjoy [11] and Corominas [4]), it follows, by Oliver [15] and also Verblunsky [18], that Dk~ 1f satisfies the Darboux property. Consequently, Dk~ 1f is continuous and non-decreasing on [a, b]. Hence, using Oliver [15] and Verblunsky [18], since Dk~1f is bounded, Dk~ if is the ordinary (k — l)th derivative/(k_1). It therefore follows, by Corollary 8 of Bullen [1], that / i s k- convex on [a, b], and thus the theorem is proved.

C orollary 2.1. Let Dk~ 1f exist on [a, b]. I f f is ACk above on [a, b]

and Dkf ^ 0 almost everywhere, then f is k-concave on [a, b].

Theorem 2.4 is now just a consequence of Theorem 2.5 and Corol­

lary 2.1.

3. The Lk-integral.

D efinition 3.1. Let / be defined on [a, b]. A continuous function M is said to be an Lk-major function of / on [a, b] if and only if

(i) Ds M exists finitely on [a, b], 1 —1;

(ii) M(a) = 1УМ(а) = 0, 1 ^ s < k - l ; (iii) M is ACk below on [a, b];

(iv) D_k M ( x ) ^ /( x ) almost everywhere in [a, b].

If — m is an Lk-major function of —f then m is called an Lk-minor func­

tion of / on [a, b].

L emma 3.1. Z/M is any Lk-major function and m is any Lk-minor function of f on [a, b], then M — m is a non-negative к-convex function on [a, b].

Proof. The proof follows from Theorem 2.5.

For a < x ^ b, define

X

F(x) = (L k) (’/ = infjr: t = M(x), M is an Lk-major function o f / )

a

as the upper Lk-integral of / on [a, x]. Similarly define

F_(x) = (Lk) I' / = sup Jr: t = m(x), m is an Lk-minor function of f \

a

as the lower Lk-integral of / on [a, x].

If F(x) = F_{x), we write the common value F(x) = (Lk) ] f

a

and further if this value is finite we say / is Lk-integrable on [a, x].

Standard arguments of Perron-type integration will give the fundamen­

tal properties of such integrals. We prove a few of them.

T heorem 3.1. I f f is Lk-integrable on [a, 6], and F is its Lk-integral, then for each e > 0 arbitrary, an Lk-major function M and an Lk-minor function m

can be chosen such that

(9) 0 ^ max {Ds (M - F) (x), Ds (F - m) (x)} < e

for all x in [a, fr] and for all s, 1 ^ s < к — 1.

Proof. The functions^M — F and F — m are 0-, 1-, 2-, ..., k-convex on [a, b f It is not difficult to show that Ds F (a) — 0 and Dk~* F (b) exists so that DS(M — F), Ds(F — m) exist on [a, b~\ for all s, 0 ^ s ^ к — 1. So, by Theorem 7(b) of Bullen [2], they are continuous on [a, b~\. Hence it is sufficient to prove (9) for a ^ x <b.

If a < c < d < b, then / is Lk-integrable on [c, d f Then to each e > 0 it is possible to choose an Lk-major function M such that 0 ^ M(b) — F(b) < e and so by Corollary 8(b) of Bullen [1]

0 ^ ( d - c ) sDs{M -F ){c) < Ae

for each s, 1 ^ s < к — 1, where A is independent of c, M — F, c. This proves (9) in [c, d f

Let fi0 = c < /?! < fi2 < • • • < b, lim Д,- = b and let Bj, j ^ 0 , be a se- quence of positive numbers to be specified later. / is L -integrable on [fijt fij+ J . Let Fj be the Lk-integral of / on [fij9 Pj+\]. To each there is an Lk-major function Mj such that 0 < Ds(Mj — Fj)(x) < for all x in [pj, fij+f]. Since in fact M j — F j is defined on [fiJf b], we have 0 < Mj — Fj < e0 on that interval for all j ^ 0. Define the functions Pj and Q} on [fijt PJ+ J , j ^ 0, inductively as follows:

k ~ ^ 1 x — В •)*

Po = 0, Qo = Po + (Mo-Fo), Pj(x) = I — — IPQj-APj),

s= 0

Q j = P j + (M j - F j ).

Characterization o f к-fold Lebesgue integral 55

Then

\D*Qo\ < 4 , 1 < s к — 1 ;

Choose fij, Ej, j ^ 0, so that |Z)S Qj\ <e, 0 ^ s ^ k — l , j ^ 0. Now define

and set M = R + F in [a, ft]. Obviously, M is an Lk-major function of / on [a, ft] such that

for all x in la, ft).

Similarly we can show that there is an Lk-minor function m such that

and the theorem is proved.

T heorem 3.2. I f f is if-integrable on [a, ft] and F is its if-integral, then (a) F is ACk on [a, ft],

(b) F(k) exists and equals f almost everywhere in [a, ft].

Proof, (a) Let e > 0 be arbitrary. By Theorem 2.1, there is an Lk-major function M such that

Since M is ACk below and M — F is к -convex on [a, ft], there is <5 > 0 such Я(х) = Qj{x), P j ^ x ^ pj + 1 ;

R{b) = limR(y),

0 ^ DS(M — F)(x) < e, O ^ s ^ k — 1

0 < DS(F — m)(x) < г, 0 ^ s < к — 1, a ^ x < ft,

< (1/ 2) (fc — 1) Ïe.

Consider any elementary system in [a, ft]

■f(Xjfi* •••> (Xj(Q, X{'k), i 1, 2, ..., n.

П

that whenever £ (xlik — xi>0) < ft we have i= 1

П

Xi,o) Qk{F > Xjo, ..., xijk)

n

i= 1

> —£/2— {Dk~ 1(M — F)(b) — Dk~ l (M — F) (a)}/(k - 1 ) !

> — e/2 —e/2 = — e.

Hence F is ACk below on [a, ft]. Similarly we can show that F is ACk above on [a, ft] and so F is ACk on [a, ft].

(b) In view of (a), Dk F exists finitely in [a, ft] — D, where mD = 0. Let e > 0 be arbitrary. By Theorem 3.1, there is an Lk-major function M such that

0 ^ D S{ M - F ) < e , 0 ^ s ^ k - l .

Then M — F is k-convex and Dk~ 1 (M — F)(b) exists. So, utilizing Theorem 17 of Bullen [1], Dk(M — F) exists finitely almost everywhere in [a, ft].

For any positive number A, define

Ex = {x: (X < x < P, a < a < ft < ft and D_k( M - F ) ( x ) ^ A}.

It follows that

Xm*Ex ^ 2 k { D k~1( M - F ) ( P ) - D k- 1(M-F)(oc)}

and so m* Ex < 2/ce/A. Consequently, m* Ex = 0. Since M is an Lk-major function of / on [a, ft], Dk M{x) ^ / (x) for all x in [a, ft] — 2s, mE = 0. Then for all x in [a, f t ] - ( £ u û u £ A) we have Dk F ( x /( x ) and so Dk F(x)

^ / (x) almost everywhere in [a, ft].

Since —/ is also Lk-integrable, we have Dk F ^ / almost everywhere in [a, ft]. As F(k) exists finitely almost everywhere, it follows that F(k) = Dk F = / almost everywhere in [a, ft] and thus (b) is proved.

We readily obtain an equivalent definition of the Lk-integral.

D efinition 3.2. A function / defined in [a, ft] will be said to be Lk~

integrable on [a, ft] if there exists a function F in [a, ft] such that (i) Ds F exists finitely on [a, ft], 1 ^ s ^ к — i ;

(ii) F (a) = DsF{a) = 0, 1 ^ s < fc-1;

(iii) F is ЛСк on [a, ft];

(iv) Dk F = f almost everywhere in [a, ft].

The function F is called the indefinite Lk-integral of /.

T heorem 3.3. Every ACk function on [a, ft] is the If-integrai o f its own k-th derivative.

Proof. Let F be ACk on [a, ft]. Then in view of Theorem 2.3 and Corollary 2.1 of De Sarkar and Das [8], F(r) exists in [a, ft], 1 < г ^ к — 1, and F(k) exists a.e. in [a, ft]. Define

<p(x ) — F(x) —

k - 1

X F ^ ( a ) - ( x - a Y . r= 0

Clearly, <p(x) is ACk on [a, ft], cp(a) = <pir)(a) = 0, —1, and <p(k)(x)

= F(k)(x) almost everywhere in [a, ft]. Hence, in view of Definition 3.2, (p is

the Lk-integral of F(k) on [a, ft].

Characterization o f к-fold Lebesgue integral

Consequently then

к — 1 x

F(x) = X F<r)(a )(x -a )r + (Lk) j F (k>,

r= 0 a

and thus the theorem is proved.

N o t e 3.1. If F eAC k [a, b] and F (a) = F(r) (a) = 0, 1, then F(x) = (Lk) f F (k).

a

T heorem 3.4. Lk-integral is equivalent to the к-fold Lebesgue integral.

Proof. If F is the indefinite Lk-integral of/ on [a, b], then, by Theorem 3.2, F is ACk on [a, b] and F(k) = / a.e. in [a, b]. By Corollary 2.3 of De Sarkar and Das [9], F(k) is summable (L) on [a, b]. Since F(r) exists everywhere in [a, b] and Fir)e ACk_r [a, b~], it follows that F is the к-fold integral of /. (Note that F(r)(a) = 0, \ ^ r ^ k — 1.)

Next, if F is the к-fold integral of / on [a, b], then, by Theorem 2.2, F is ACk on [a, b] and F(k) — f a.e. in [a, b]. Clearly, F(,) exists in [a, b] and F(r)(a) = 0, 1 < r < fc —1. Thus F is both an Lk-major and Lk-minor function of / on [a, b], and hence is the Lk-integral of / on [a, b]. This proves the theorem.

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DEPARTMENT O F MATHEMATICS, UNIVERSITY O F KALYANI KALYANI, NADIA, WEST BENGAL