p*****n
发帖数: 758 | 1 suppose (X,M,u) is a measure space, F is a family of non-negative integrable
functions on X with the following properties:
1. if f,g are in F, so is f+g.
2. if f,g are in F, so is max{f,g}.
3.if f is a non-negative measurable function with positive integration, then
there exists a function g in F, such that g<=f, g has postive integration.
4.zero function is in F.
prove that for any non-negative measurable function f,integration of f=sup{
integration of g: g is in F, g<=f}.
thx | l*****e
发帖数: 65 | 2 用反证法试试.
1. Assume that there exists a non-negative measurable function f, such that
the supremum{ integration of g: g \in F and g <= f} is strictly less than
the integration of f. Denote the difference by \delta>0.
2. By the definition of the supremum, there exists a sequence of non-
negative integrable functions g_i, g_i \in F and g_i <=f, such that \int{f-g
_n} <= \delta + 1/n.
3. Define a new sequence of non-negative integrable
functions G_i, such that
G_1=g_1, and G_n=max(g_n, G_{n-1}) for n | p*****n
发帖数: 758 | 3 谢谢 ！
that
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【在 l*****e 的大作中提到】
: 用反证法试试.
: 1. Assume that there exists a non-negative measurable function f, such that
: the supremum{ integration of g: g \in F and g <= f} is strictly less than
: the integration of f. Denote the difference by \delta>0.
: 2. By the definition of the supremum, there exists a sequence of non-
: negative integrable functions g_i, g_i \in F and g_i <=f, such that \int{f-g
: _n} <= \delta + 1/n.
: 3. Define a new sequence of non-negative integrable
: functions G_i, such that
: G_1=g_1, and G_n=max(g_n, G_{n-1}) for n
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