L p Functions Given a measure space (, µ) and a real number p [, ), recall that the L p -norm of a measurable function f : R is defined by f p = ( ) /p f p dµ Note that the L p -norm of a function f may be either finite or infinite. functions are those for which the p-norm is finite. The L p Definition: L p Function Let (, µ) be a measure space, and let p [, ). An L p measurable function f on for which f p dµ <. function on is a Like any measurable function, and L p function is allowed to take values of ±. However, it follows from the definition of an L p function that it must take finite values almost everywhere, so there is not harm in restricting to L p functions R. It is easy to see that any scalar multiple of an L p is again L p. Moreover, if f and g are L p functions, then by Minkowski s inequality f + g p f p + g p < so f + g is an L p function. Thus the set of L p functions forms a vector space. EAMPLE L p Functions on [0, ] Any bounded function on [0, ] is automatically L p for every value of p. However it is possible for the p-norm of a measurable function on [0, ] to be infinite. For example,
2 let f : [0, ] R be the function f(x) = x where the value of f(0) is immaterial. Then by the monotone convergence theorem, [ ] f dm = lim dm(x) = lim log x a 0 + x = a 0 + a [0,] [a,] so f is not L. Indeed, it is easy to check that f is not L p for any p [, ). A function with a vertical asymptote does not automatically have infinite p-norm. For example, if f(x) = x then f has a vertical asymptote at x = 0, but [ f dm = lim dm(x) = lim 2 x a 0 + x a 0 + [0,] [a,] ] a = 2. In general, [0,] x r dm(x) = if r /( r) if r <. It follows that the function f(x) = /x r is L p if and only if pr <, i.e. if and only if p < /r. For example, f(x) = / x is L p for all p [, 2), but is not L p for any p [2, ). The last example suggests that it should be harder for a function to be L p the larger we make p. The following proposition confirms this intuition. Proposition Relation Between L p and L q Let (, µ) be a measure space, and let p q <. If µ() =, then f p f q for every measurable function f. More generally, if 0 < µ() <, then f p µ() r f q for every measurable function f, where r = (/p) (/q), and hence every L q function is also L p.
3 PROOF The case where µ() = is the generalized mean inequality for the p-mean and the q-mean. For 0 < µ() <, let C = µ(), and let ν be the measure dν = C dµ. Then ν() =, so by the generalized mean inequality ( ) /p ( ) /p f p dµ = C /p f p dν ( ) /q ( /q C /p f q dν = C /p C /q f dµ) q. Note that this proposition only applies in the case where µ() is finite. As the following example shows, the relationship between L p and L q functions can be more complicated when µ() =. EAMPLE 2 Horizontal Asymptotes Let f : [, ) R be the function f(x) = x. Then f is not L, since by the monotone convergence theorem [ ] b f dm = lim dm(x) = lim log x [, ) b [,b] x =. b However f is L 2, since In general, [, ) f 2 dm = lim b [,b] [, ) x r dm(x) = [ dm(x) = lim ] b x2 b x /(r ) if r > if r. =. Thus f(x) = /x r is L p if and only if pr >, i.e. if and only if p > /r. Thus, for horizontal asymptotes it is easier for a function to be L p the larger the value of p. Intuitively, this is because numbers close to 0 get smaller when taken to a larger power, so f p will be closer to the x-axis the larger the value of p.
4 l p Sequences An important special case of L p functions is for the measure space (N, µ), where µ is counting measure on N. In this case, a measurable function f on N is just a sequence f(), f(2), f(3),... and the Lebesgue integral is the same as the sum of the series f dµ = f(n). N The definition of an L p function on N takes the following form. Definition: l p -Norm and l p Sequences If p [, ), the l p -norm of a sequence {a n } of real numbers is defined by the formula ( ) /p {a n } p = a n p. An l p sequence is a sequence {a n } of real numbers for which a n p <. Sequences behave in a similar manner to functions with horizontal asymptotes. EAMPLE 3 P -series Recall that the p-series converges if and only if p >. It follows that the sequence {/n p } is l if and only if p >. For example, { } { } { } is l but and n are not. n 2 n Moreover, since (/n r ) p = /n rp, we find that {/n r } is l p if and only if p > /r. Thus { } is l 2 but not l, n n p
5 and { n } is l 3 but not l 2. All of this is very similar to our analysis of the function /x p on [, ]. Indeed, it follows from the integral test that x dx < if and only if p n p < so there is a strong theoretical relationship between these two cases. Proposition 2 Relationship Between l p and l q If p < q <, then every l p sequence is also l q. PROOF Let {a n } be an l p sequence. Then a n p converges, so it must be the case that a n 0 as n. In particular, there exists an N N such that a n < for all n N. Then a n q < a n p for all n N, so a n q converges by the comparison test. Incidentally, Hölder s inequality is very interesting for sequences, since it essentially functions as a new convergence test for series.
6 Theorem 3 Hölder s Inequality for Sequences Let {a n } and {b n } be sequences of real numbers, and let p, q [, ) so that /p + /q =. If the series a n p and b n p both converge, then the series converges absolutely, and a n b n a n b n ( ) /p ( ) /q a n p b n q. Corollary 4 Cauchy-Schwarz Inequality for Sequences Let {a n } and {b n } be sequences of real numbers. If the series a 2 n and b 2 n both converge, then the series converges absolutely, and a n b n ( ) 2 ( a n b n ) ( a 2 n b 2 n ).
7 L p Completeness It is possible to generalize the completeness theorem to L p. Definition: L p Sequences Let (, µ) be a measure space, let {f n } be a sequence of measurable functions on, and let p [, ).. We say that {f n } is an L p Cauchy sequence if for every ɛ > 0 there exists an N N so that i, j N f i f j p < ɛ. 2. We say that {f n } has bounded L p -variation if f n+ f n p <. 3. We say that {f n } converges in L p to a measurable function f if lim f n f p = 0. n Theorem 5 L p Convergence Criterion Let (, µ) be a measure space, and let {f n } be a sequence of measurable functions on with bounded L p -variation. Then {f n } converges pointwise almost everywhere to a measurable function f, and f n f in L p. PROOF Let M = f n+ f n p <. and let N g = f n+ f n and g N = f n+ f n for each N N. By Minkowski s inequality, g N p N f n+ f n p M
8 for all N N. By the monotone convergence theorem, it follows that g p dµ = lim N gp N dµ = lim g p N dµ = lim g N p p M p <. N N From this we conclude that g(x) < for almost all x, so {f n (x)} has bounded variation for almost all x, and hence {f n (x)} converges pointwise almost everywhere. Let f be the pointwise limit of the sequence {f n }, and note that for each n N, f f n = lim f N+ f n = N lim N k=n N (f k+ f k ) = (f k+ f k ) k=n almost everywhere. Then f f n p ( ) p = fk+ f k k=n ( ) p f k+ f k g p k=n almost everywhere, so by the dominated convergence theorem f f n p dµ = lim f f n p dµ = 0. n Thus f n f in L p. lim n L p completeness follows easily. We leave the proof to the reader. Theorem 6 L p Completeness Let (, µ) be a measure space, and let {f n } be an L p Cauchy sequence on. Then {f n } converges in L p to some measurable function f on.
9 The L Norm It is possible to extend the L p norms in a natural way to the case p =. Definition: L -Norm Let (, µ) be a measure space, and let f be a measurable function on. The L -norm of f is defined as follows: f = min { M [0, ] f M almost everywhere }. We say that f is an L function if f <. Note that the set { M [0, ] f M almost everywhere } really does have a minimum element, for if f M + /n almost everywhere for all n N, then it follows that f M almost everywhere. The L -norm f is sometimes called the essential supremum of f, and L functions are sometimes said to be essentially bounded or bounded almost everywhere. Note that a continuous function on R is L if and only if it is bounded, in which case f is equal to the supremum of f. Much of what we have done for p [, ) also works for p =. We list some of the results, and leave the proofs to the reader: Minkowski s Inequality. If f and g are L functions, then f + g is L, and f + g f + g. Hölder s Inequality. If f is an L function and g is an L function, then fg is Lebesgue integrable and f, g f g. L Convergence. If {f n } is a sequence of functions, we say that {f n } converges in L to a function f if lim f n f = 0. n This turns out to be the same as uniform convergence almost everywhere, i.e. f n f in L if and only if there exists a set Z of measure zero such that f n f uniformly on Z c.
0 L Completeness. If {f n } is an L Cauchy sequence of measurable functions, then {f n } converges in L to some measurable function f. Relation Between L and L p If µ() =, then f p f for any measurable function f on. More generally, if 0 < µ() < then f p µ() /p f for all p, so any L function on is also L p for all p [, ). In the case of sequences, the L norm takes the following form. Definition: l -Norm Let {a n } be a sequence of real numbers. The l -norm of {a n } is defined as follows: {a n } = sup a n Thus an l sequence is the same as a bounded sequence. Note that if p [, ), then any l p sequence must be l, since any l p sequence must converge to zero. Exercises For the following exercises, let (, µ) be a measure space.. Let f : [0, ) R be the function f(x) = e x. For what values of p is f an L p function? 2. Let f : (0, ) R be the function f(x) = { x /3 0 < x <, x /2 x <. For what values of p is f an L p function? 3. Let f : [0, ] [0, ] be the function f(x) = log x, with f(0) =. (a) Show that f is L. (b) Show that f is L p for all p [, ). (Hint: Substitute u = /x.)
4. For what values of p is { } (n 2 + ) /3 an l p sequence? 5. For what values of p is { } n log n an l p sequence? 6. Prove that every L p Cauchy sequence has a subsequence of bounded L p -variation. 7. Prove the L p completeness theorem (Theorem 6). 8. If f and g are measurable functions on, prove that f +g f + g. 9. If f is an L function on and g is an L function on, prove that fg is Lebesgue integrable and f, g f g. 0. Let {f n } be a sequence of measurable functions on, and let f be a measurable function on. Prove that f n f in L if and only if f n f uniformly almost everywhere.. If 0 < µ() < and f is a measurable function on, prove that for all p [, ). f p < µ() /p f 2. Prove that every L Cauchy sequence of measurable functions converges uniformly almost everywhere.