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Extra info for 1973, Year of the Humanoids: An Analysis of the Fall UFO Humanoid Wave

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1 The next equality is trivial and the final inclusion holds because of the selection process fop the sets Sk. So we have h I^ -U%le ~ ~ 1 Making 9nn ~ e we obtain the result. j=h+1 9nlsjl ~ 9nn 23 If A is not inside Q we apply the result we have obtained to the intersection of A with the interior of each one of the cubes of Rn with side-length I and vertices at the points with integral coordinates. The proof we have presented lends itself to interesting generalizations that we shall present later on in the final remarks of this section.

Since H is arbitrary we obtain it for f at almost every point. This observation permits us t o prove a result a little finer. Le! f e Lloc(Rn). Then, for almost every x e Rn, we have l l m ~ r+O If If(y) - f(x)Idy = O. Q(x,r) To prove this, let {qk}~:l be an enumeration of all rational numbers. ) - qkl, which is obviously in Lloc(Rn). Then if for each k we exclude a nullset Ek, we have, for each x g E k ~ l ~ mi ~+0 IQ(x,r)If(Y) - qkldy : If(x) - qkl" Let E = 0 E k" Obviously IEI = 0 and if z ~ E, then, for each K k=l r~+O ~ l l m1 IQ(z,r)If(Y) - qkldy = If(z) - qkl Let A = {.

F e Lloc(Rn). Then, for almost every x e Rn, we have l l m ~ r+O If If(y) - f(x)Idy = O. Q(x,r) To prove this, let {qk}~:l be an enumeration of all rational numbers. ) - qkl, which is obviously in Lloc(Rn). Then if for each k we exclude a nullset Ek, we have, for each x g E k ~ l ~ mi ~+0 IQ(x,r)If(Y) - qkldy : If(x) - qkl" Let E = 0 E k" Obviously IEI = 0 and if z ~ E, then, for each K k=l r~+O ~ l l m1 IQ(z,r)If(Y) - qkldy = If(z) - qkl Let A = {. e ~n : If(x)l : | Clearly IAI : O. Take now a fixed v ~ A U E and e > O.

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