Zero Slice Theorem

Let G graph f foreveryXER s f X E112 thensince is afunction exfax I at mostone year s t Y fax Therefore a slice Therefore m Gx Gxof G has at mostonepoint 0 SliceTheorem then m G 0 Zero the By Let E C R be set and let fix a nonmeasurable Then exfax XER MI R x 04 t m Rx megraph f Rx C U Rx I 0 0 0 O I Ie x I thensince 0 But E nonmeasurable I f nonmeasurable so thestatementdoesnothold shouldbesince Ex O I unmeasurable sketchof construction EnumerateGgsets in f R Let f R R H a IN as Aafor a I f N an approximation to f st is definedonlyon a points If we can addonemorepoint to thedomainoffa s t is outside Ha thendo so stackexchange subject If not x fat x1 Ancomainstheonptened ii Lebesgue Measure ofthe Graph of why I want to draw a picture Perhaps this 11111in 1111 11111 1111 Eventually of thisstep 7 but notsure how 11 page 11 I if we extendthefunction to f 111 Thegraph off is not anyGssetwith no full measure graphof has full intermeasure Howdoesthis lead to contained in In Theorem 26 for a measurable set E C R xR if almost all slicesof E is a zero set E is a zero set Thus M E If O M E O we dropthemeasurability requirement then we constructed in cc Funtionable forthe since eachslice of its graph has measure zero However thegraph of thefunction has positive outer measure is a point in R it graph f Let G Thensince f is a function everyslice Gx of G contains onlyonepoint thushasmeasurezero F C G its slices Fx willalsoonlycontain onepoint So bythe Zen SliceTheorem McG O Therefore G cannothave positiveinnermeasure Now consider Thefunction f measurablesubset any in co is f La b o o a positive outermeasure Then define Then a b x R c refgraph g R c Lggraph gin tr re R I think If we extend the domain off to R functions gr then graph gr c graph gi Need to show m graphight n Then gutsfix whosegraphhas to construct i Idea 19 showsthat I From Cf a meseometry t we can send a subset of preserving measurability ofgraph f I denoted algorithmthatsends I R to R denoted E takesubsets ofthe translations while Fi withthepre imageequal to E Thenthis E to graph f is themeseometry Thepositiveandnegativepartsofthefunction is f f x max fix O v maxi Let Et E xy XER y xy X ER Yco Weknow Et E find t ggp 0 Y E measurable f yet is measurable then u ft measurable ft measurable Ucfi n E Assume E u ft s u f s ft f a ft U Assume Alfis Ucf measurable Ucf OE anemeasurablethen u Cucf nE ft ucf f measurable Ucf measurablethen measurable We know Ucf is measurable and want u tf to show is measurable From Ex 23 we knowthat a diffeomorphism is a meseomorphism Since T thuspreserves measurability Ucf ut is a diffeomorphism and ex y to ex s ucf is measurable is measurable ult is measurable so tf First it is clearthat both T and T are diffeomorphisms sotheybothpreserve measure Then f g measurable acts u g measurable Ullogf Next we have yy uclogg measurable logf logg measurable lost log9 measurable logfix t loggox log fix go logf t logg s logifg uclogftlogg s ullogfg measurable Since T preserves measureand sends uclogfg to actg Ucfg is measurable thus fg is measurable For ca we caneasily generalize to R since theset operations hold in higherdimensions ForCb and c b T Xi c T Xi we can finddiffeomorphism T in higherdimensions xn y Xn y t h Xr Xn ti Xi Xn logy For f g that have bothsigns weonlyneed to divide we Ucf and ug bythe 4 quadrants Then can show that individually the 4 quadrants are Thenthe union ofthe pieces will be measurable measurable