Mapping the delta function and other Radon measures


 Melvyn Bryant
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1 Mpping the delt function nd other Rdon mesures Notes for Mth583A, Fll 2008 November 25, 2008 Rdon mesures Consider continuous function f on the rel line with sclr vlues. It is sid to hve bounded support if there is bounded intervl [, b] of rel numbers such tht f(x) 0 implies x b. An lterntive terminology tht is often used is tht f is sid to hve compct support. Let C c denote the vector spce of continuous sclr vlued functions on the rel line, ech of which hs compct support. This will be clled the spce of (continuous) test functions. A Rdon mesure is positivity preserving liner function µ from C c to the sclrs. The vlue of the Rdon mesure µ on the test function f is denoted µ, f. The positivity preserving condition sys tht if for ll x we hve f(x) 0, then µ, f 0. Technicl note: In mesure theory Rdon mesure would generte mesure defined on the Borel subsets of the line tht is finite on compct subsets. Exmple. Let h 0 be positive loclly integrble function. This mens tht for ech point p there is constnt c > 0 such tht the integrl p+c p c h(x) dx <. () It follows tht for every bounded intervl [, b] of rel numbers we hve Then there is Rdon mesure µ h given by µ h, f = h(x) dx <. (2) h(x)f(x) dx. (3) Exmple 2. Let be rel number. The point mss t is the Rdon mesure δ defined by δ, f = f(). (4)
2 One cn lso tke liner combintions j c jδ j with c j 0. The vlue of this on function is of course j c j δ j f = j c j f( j ) (5) The intuitive interprettion of Rdon mesure is s s mss spred out on the line. The most generl Rdon mesure is defined s follows. Let F be n incresing rightcontinuous function. To sy tht F is incresing is to sy tht x y implies F (x) F (y). To sy tht F is right continuous is to sy tht lim ɛ 0 F (x + ɛ) = F (x). Then the most generl Rdon mesure is given by the RiemnnStieltjes integrl µ F, f = f(x) df (x). (6) Given the Rdon mesure, the function F is determined up to constnt of integrtion. The prt corresponding to the jump discontinuities of F is the prt corresponding to point msses. The prt given by density function h 0 is clled the bsolutely continuous prt. For this prt the function F (x) = h(x) lmost everywhere, nd, furthermore, the function F my be recovered from h s n indefinite integrl. (There cn be third prt to the mesure clled the singulr continuous prt.) 2 Signed Rdon mesures One cn lso define signed Rdon mesures. The simplest definition is to tke this s the difference of two Rdon mesures. (In similr wy, one cn define complex Rdon mesures.) Technicl note: A signed Rdon mesure is not necessrily signed mesure in the sense of mesure theory, since it could hve both infinite positive prt nd infinite negtive prt. Exmple. Let h be loclly integrble function. This mens tht for ech point p there is constnt c > 0 such tht the integrl p+c p c h(x) dx <. (7) It follows tht for every bounded intervl [, b] of rel numbers we hve Then there is signed Rdon mesure µ h given by µ h, f = h(x) dx <. (8) h(x)f(x) dx. (9) 2
3 Exmple 2. Let be rel number. The point mss t is the Rdon mesure δ defined by δ, f = f(). (0) One cn lso tke liner combintions j c jδ j with c j rel. The vlue of this on function is of course j c j δ j f = j c j f( j ) () The intuitive interprettion of signed Rdon mesure is s s chrge spred out on the line. 3 Schwrtz distributions Schwrtz distributions re more generl thn signed Rdon mesures. For Schwrtz distributions the test functions re restricted to be in Cc, the spce of smooth functions ech of which hve compct support. Ech Schwrtz distribution T defines liner function T from Cc to the sclrs. The vlue of T on f is denoted T, f. Exmple. The principl vlue /x integrl is defined by P V x, f = lim x ɛ 0 x 2 f(x) dx. (2) + ɛ2 Exmple 2. The derivtive of the point mss t is defined by δ, f = f (). (3) The importnt new feture is tht Schwrtz distribution need not hve decomposition into positive prt nd negtive prt. In fct, the only time when this decomposition exists is when the Schwrtz distribution is ctully signed Rdon mesure. 4 Functions nturlly mp bckwrd Functions re nturlly covrint nd mp bckwrd. Sy tht u = φ(x). If g(u) is function, then the function g(φ(x)) is the pullbck. This is the nturl opertion on functions. Similrly, if g(u) du is differentil form, then g(φ(x))φ (x) dx is the nturl covrint pullbck. Then we hve tht g(φ(x))φ (x) dx = φ(b) φ() g(u) du. (4) Here is consequence of this eqution tht will be importnt in the following. Suppose tht φ(x) is either incresing or decresing on the intervl (, b) with 3
4 < b. Then pplying the bove formul g(φ(x)) replced by h(φ(x))f(x) we get φ(b) h(φ(x))f(x) dx = h(u)f(φ (u)) φ (φ du. (5) (u)) φ() Now brek up the rel xis into such intervls (, b). On those intervls where φ(x) is decresing we interchnge φ() nd φ(b) nd replce φ (x) by φ (x). Since there cn be severl intervls on which φ(x) = u, the finl result is tht h(φ(x))f(x) dx = h(u)[ f(x) φ ] du. (6) (x) It is possible in prticulr tht there my be no x with g(x) = u, in which cse we interpret the sum s zero. 5 Functions cn mp forwrd Sometimes function is tken to mp forwrd. In tht cse it is often cllled density. Thus densities re contrvrint nd mp forwrd. The function ρ(x) might represent probbility density or mss density or chrge density. Sy tht φ(x) is function whose derivtive only vnishes t isolted points. The expecttion of g(φ(x)) with respect to this density is g(φ(x))ρ(x) dx = Thus the pushforwrd of ρ(x) is φ [ρ](u) = g(u)[ ρ(x) φ ] du. (7) (x) ρ(x) φ (x). (8) So when function is interpreted s density, it is contrvrint object. 6 Signed Rdon mesures nturlly mp forwrd A signed Rdon mesure ssigns to ech continuous function f with compct support number µ, f. Signed Rdon mesures re nturlly contrvrint nd mp forwrd. If g φ is the composite function defined by (g φ)(x) = g(φ(x)), then the pushforwrd mesure φ [µ] is defined by φ [µ], g = µ, g φ. Thus, for instnce, if the mesure is given by density ρ(x), then the push forwrd of µ is given by ρ(x)/ φ (x). 4
5 Similrly, the pushforwrd of δ is δ φ(). Here is n interesting exmple. Sy tht φ(x) = is constnt function. Then the pushforwrd of mesure with density ρ(x) is the mesure ρ(x) dx δ. So mesure given with density goes into point mesure. 7 Signed Rdon mesures cn lso mp bckwrd A signed Rdon mesure (or more generlly, Schwrtz distribution) cn lso be interpreted s generlized function. Generlized functions re covrint nd mp bckwrd. It is obvious how to do this with density: the density ρ(u) is mpped to the density ρ(φ(x)). Notice tht this my not preserve the totl mss or the totl chrge. The generl rule is the pullbck of the signed Rdon mesure µ is the signed Rdon mesure φ [µ] defined by φ [µ], f = µ, φ [f]. (9) Exmple. A point mss my be mpped bckwrd. The generl formul is tht φ [δ b ], f = φ () δ. (20) φ()=b The motivtion for this formul comes if we write the point mss s if it were function. Thus we define n object δ(u b) tht is relly point mss, so Then the pullbck is determined by This is equl to δ(φ(x) b)f(x) dx = φ()=b f() φ () = δ(u b)g(u) du = g(b). (2) [ φ()=b In other words, the pullbck of δ(u b) is δ(φ(x) b) = φ()=b δ(u b)[ φ()=u f() φ ] du. (22) () δ(x ) φ ]f(x) dx. (23) () δ(x ) φ (). (24) Schwrtz distributions re generliztion of signed Rdon mesures. Distributions nturlly mp forwrd, but ordinrily they re considered s generlized functions, so in prctice we usully do the unnturl thing nd mp them bckwrds. Such is life. 5
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