Math ASCII Notation Demo

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Let F be a s-algebra in X.
(1) f : X -> |R*.
(2) n:-|N
(3) a[1], ..., a[n] :- |R
(4) A[1], ..., A[n] :- F , some of them may be empty
(5) X = \\*//(k=1 to n) A[k]
(6) /\x:-X f(x) = +(k=1 to n) a[k]*1(A[k])(x)
f is simple and F-measurable
page 109 in 2nd measure
Let F be a s-algebra in X.
Let f : X -> [0,oo] be F-measurable.
In this setting, how can we interestingly approximate function f ?
(In this item, prove the existence of this interesting approximation.)
There exists a sequence of functions { s[n] }(n:-|N) such that
(1) /\n:-|N s[n] : X -> [0,oo) is simple and measurable
(2) /\x:-X /\n:-|N s[n](x) <= s[n+1](x)
(3) /\x:-X lim(n->oo) s[n](x) = f(x)
page 113 in 2nd measure
f : X -> [0,oo]
In this broad setting, how can we interestingly approximate function f ?
(In this item, do not prove the existence of this interesting approximation.)
There exists a sequence of functions { s[n] }(n:-|N) such that
(1) /\n:-|N s[n] : X -> [0,oo) is simple
(2) /\x:-X /\n:-|N s[n](x) <= s[n+1](x)
(3) /\x:-X lim(n->oo) s[n](x) = f(x)
page 110 in 2nd measure
Let f : X -> [0,M] , M < oo.
In this broad setting, how can we interestingly approximate function f ?
There exists a sequence of functions { s[n] }(n:-|N) such that
(1) /\n:-|N s[n] : X -> [0,oo) is simple and bounded
(2) /\x:-X /\n:-|N s[n](x) <= s[n+1](x)
(3) /\x:-X lim(n->oo) s[n](x) = f(x)
(4) sequence s[n] converges uniformly to function f
page 113 in 2nd measure
======================
lim(x,y)->(0,0) y*(x-sin(x)) / (x^4 + y^2) = ???
lim(x,y)->(0,0) y*(x-sin(x)) / (x^4 + y^2) = 0
hint:
x-sin(x) = x^3 * u(x), where lim(x->0) u(x) = 1/6
|xy/(x^2+y^2)|<=1/2
page 100 in OLDTIMER
Let a : |N -> |C.
Let z be a complex number.
Suppose that series a[n]*|z|^n converges.
Can we conclude that series a[n]*z^n converges?
NO
(1) Let a[n] = 1/n * (-1)^n. Let z = (-1).
(2) Consider series (-1)^n / n * z^(3n). Let z = -1.
(1) W is a ring of sets in X
(2) y : W -> [0,oo) is additive
(3) A[1] , ... , A[n] :- W (disjoint sets)
(5) B[1] , ... , B[m] :- W (disjoint sets)
(6) X = U(k=1 to n)_A[k] = U(j=1 to m)_B[j]
(7) a[1], ... , a[n] , b[1] , ... , b[m] are positive real numbers
+(k=1 to n) a[k]*y(A[k]) = +(j=1 to m) b[j]*y(B[j])
page 147 in 1st measure
Let W be a s-algebra in X.
Let y : W -> [0,oo] be additive, y(O) = 0.
Let f : X -> [0,oo) be a simple and W-measurable function.
Define the Lebesgue integral of function f with respect to y.
LebS(X,f,dy) = ???
Since f is simple and W-measurable, there exist:
(1) disjoint sets A[1], A[2], ..., A[n] :- W, with X = U(k=1 to n) A[k]
(2) a[1], a[2], ..., a[n] :- [0,oo)
such that (5) /\x:-X f(x) = +(k=1 to n) a[k]*1(A[k])
Define LebS(X,f,dy) := +(k=1 to n) a[k]*y(A[k]).
Let W be a s-algebra in X.
Let y : W -> |R* be additive.
Let A :- W.
LebS(X,1(A),dy) = ???
LebS(X,1(A),dy) = y(A)
page 149 in 1st measure
Let W be a s-algebra in X.
Let f,g : X -> |R be simple and W-measurable.
Prove that there exist:
(1) E[1], ... , E[n] :- W (disjoint, having union X)
(2) a[1], ... , a[n], b[1], ... , b[n] :- |R
such that
(3) f = +(k=1 to n) a[k]*1(E[k])
page 149 in 1st measure
Notice the technical difficulty here.
One day I want to go deeper into this theorem.
MRW 2001.02.24; 2001.06.02