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f ( x , y ) x ∈ R d 1 , y ∈ R d 2 2010.05.26 F u b i n i T h m _ S u p p o s e f ( x , y ) i s i n t e g r a b l e , t h e n t h e f o l l o w i n g h o l d : ( i ) f i ∈ J , i = 1 , 2 , ⋯ , m Σ a i f i ∈ J ( i i ) f n ∈ J , e i t h e r f n ( x ) ≤ f n + 1 ( x ) , o r f n ( x ) ≥ f n + 1 ( x ) , t h e n f ( x ) ∈ J . P r o o f _ A s s u m e ( i ) & ( i i ) h o l d s . W a n t t o p r o v e J = { i n t e g r a b l e f u n c t i o n } f : { i n t e g r a b l e } = f + − f − , f + a n d f − ∈ J . B y ( i ) , f ∈ J . f ≥ 0 , ∃ f n s i m p l e f u n c t i o n b e l o n g t o J , t h e n b y ( i i ) , f ∈ J . S i n c e a s i m p l e f u n c t i o n i s a s u m m a t i o n o f c h a r a c t e r i s t i c f u n c t i o n s , ∑ i = 1 N a j χ E j , E j : m e s u r a b l e a n d f i n i t e m e a s u r e . ( ∵ ∑ i = 1 N a j χ E j i s i n t e g r a b l e ) . I f χ E ∈ J f o r a n y m e a s u r a b l e s e t E w i t h m ( E ) < + ∞ t h e n a n y s i m p l e f u n c t i o n b e l o n g s t o J . D o e s χ E ∈ J ? E = G δ ∪ F δ m ( F σ ) = 0 χ E = χ G δ + χ F σ I f χ G δ ∈ J , χ F σ ∈ J , t h e n χ E ∈ J . ( 1 ) F σ m e a s u r e z e r o ? ⇒ χ F σ ∈ J ( 2 ) G δ = ∪ n = 1 ∞ O n ⇒ χ G δ ∈ J {\displaystyle {\begin{aligned}&f(x,y)\;x\in \mathbb {R} ^{d_{1}},y\in \mathbb {R} ^{d_{2}}\;2010.05.26\\&\mathrm {{\underline {Fubini\;Thm}}\;Suppose\;} f(x,y)\mathrm {\;is\;integrable,\;then\;the\;following\;hold:} \\&(i)f_{i}\in {J},\;i\mathrm {=} 1,2,\cdots ,m\;\Sigma {a}_{i}f_{i}\in {J}\\&(ii)f_{n}\in {J}\mathrm {,\;either\;} f_{n}(x)\leq {f}_{n+1}(x)\mathrm {,\;or\;} f_{n}(x)\geq {f}_{n+1}(x)\mathrm {,\;then\;} f(x)\in {J}.\\&\mathrm {{\underline {Proof}}\;Assume\;(i)\And (ii)\;holds.\;Want\;to\;prove\;} J\mathrm {=} \left\{\mathrm {integrable\;function} \right\}\\&f:\left\{\mathrm {integrable} \right\}\mathrm {=} f^{+}-f^{-},\;f^{+}\mathrm {\;and\;} f^{-}\in {J}\mathrm {.\;By\;(i),\;} f\in {J}.\\&f\geq {0},\;\exists {f_{n}}\mathrm {\;simple\;function\;belong\;to\;} J\mathrm {,\;then\;by\;(ii),\;} f\in {J}.\\&\mathrm {Since\;a\;simple\;function\;is\;a\;summation\;of\;characteristic\;functions,} \\&\sum _{i=1}^{N}a_{j}\chi _{E_{j}},\;E_{j}\mathrm {:mesurable\;and\;finite\;measure.(} \because \sum _{i=1}^{N}a_{j}\chi _{E_{j}}\mathrm {\;is\;integrable).} \\&\mathrm {If\;} \chi _{E}\in {J}\mathrm {\;for\;any\;measurable\;set\;} E\mathrm {\;with\;} m(E)<+\infty \\&\mathrm {then\;any\;simple\;function\;belongs\;to\;} J.\mathrm {\;Does\;} \chi _{E}\in {J}?\\&E\mathrm {=} G_{\delta }\cup {F}_{\delta }\;m(F_{\sigma })\mathrm {=0\;} \chi _{E}\mathrm {=} \chi _{G_{\delta }}\mathrm {+} \chi _{F_{\sigma }}\mathrm {\;If\;} \chi _{G_{\delta }}\in {J},\chi _{F_{\sigma }}\in {J}\mathrm {,\;then\;} \chi _{E}\in {J}.\\&\mathrm {(1)} F_{\sigma }\mathrm {\;measure\;zero} ?\Rightarrow \chi _{F_{\sigma }}\in {J}\mathrm {(2)} G_{\delta }\mathrm {=} \cup _{n=1}^{\infty }O_{n}\Rightarrow \chi _{G_{\delta }}\in {J}\end{aligned}}}
