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b.solns.billingsley

b.solns.billingsley - Probability II MAP6473 4810 Prof JLF...

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Probability II MAP6473 4810 Solns-B Prof. JLF King 4May2010 Tools. For each “speed” s R , use E s as a name for the map z 7→ e i sz on R . Note that | e i θ - 1 | 6 | θ | , for all real θ . This since chord 6 arc. 1 : By the way, unmarked integrals Z shall mean Z R . 2 : Equi-cts Lemma. On compact metric space J we have real-valued functions h n g pointwise . The convergence will be uniform , if { h n } n is a uniformly equi-continuous family. Proof. The limit g automatically has the same ε, δ -relation as the family; so we may replace each h n by h n - g . I.e, WLOGenerality The ( h n ) n converge pointwise to zero . Given ε , take the corresponding δ from the family. Now pick a δ -dense set, F , of points in J ; we can take F finite , since J is compact. Discarding the first few h n , we now have n, t F : | h n ( t ) | 6 ε ; this, by the convergence to zero. For an arbitrary point s J there is a point t F which is δ -close to s . Their h n -values are thus ε -close. The upshot is that for all n : | h n ( s ) | 6 2 ε . Consequently, back in our original notation we have that k h n - g k sup 6 2 ε , for all large n . 3 : Bnd Lemma. On prob.space , μ ) , meas.map f R has f () 6 7 . If R f d μ > 7 then f () a.e = 7 . Proof. As { f < 7 } = S j { f 6 7 - 1 j } , I need but fix a number L< 7 and show that v := μ ( f () 6 L ) is zero. But R f 6 [1 - v ] · 7 + v · L = 7 - v [7 - L ]. So v = 0. B1 : For a prob.meas. μ , let ˜ μ be “ μ flipped”; so ˜ μ ( B ) := μ ( - B ) . Write the char.fnc Φ ˜ μ ITOf ( In Terms Of ) Φ μ . Let h μ ; 7 i and h μ ; 7 , 3 i be a translation and a translation-scaling of μ : h μ ; 7 i ( B ) := μ ( B - 7) ; h μ ; 7 , 3 i ( B ) := μ (3 B - 7) . Describe the char.fncs Φ h μ ;7 i and Φ h μ ;7 , 3 i ITOf Φ μ . Soln- B1: Let Φ mean Φ μ . The problem discusses affine maps of R . Lets broaden our view to a general measurable map Q : R and define the “push forward measure” h μ ; Q i ( B ) := μ Q 1 ( B ) . Written with the in- dicator fnc, R 1 B

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