Differentiation

The Lebesgue-Radon-Nikodym theorem

Let `(Omega, F)` be a measurable space and let `mu` and `nu` be two measures on `(Omega, F)`. `mu` is dominated by `nu`, `mu << nu` if `nu(A) = 0 => mu(A) = 0 AA A in F`.

Let `f` be a non-negative measurable function, `mu(A) = int_A f dnu AA A in F`, then `mu` is also a measure and `mu << nu`.

`mu` is singular wrt to `nu`, `mu _|_ nu` if `EE B in F "st" mu(B) = 0 and v(B^c) = 0`. If `mu` and `nu` are mutally singular, then for `B` as above `mu(A) = mu(A nn B^c)` and `nu(A) = nu(A nn B)`.

• Lebesgue decomposition theorem: `mu` can be uniquely decomposed as `mu = mu_a + mu_s` where `mu_a << nu` and `mu_s _|_ nu` where `mu_a` and `mu_s` a `sigma`-finite.
• Radon Nikodym theorem: there exists a non-negative measurable function `h` on `(Omega, F)` st. `mu_s(A) = int_A h dnu AA A in F`

Let `h` be a non-negative measurable function st `mu(A) = int_A h dnu`, then `h` is called the Radon-Nikodym derivative` of `mu` wrt `nu` and is written `dmu/dnu`.

Let `nu, mu_1, mu_2, ...` be `sigma`-finite measures, then:

• `mu_1 << mu_2, mu_2 << mu_3`, then `mu_1 << mu_3` and `dmu_1/dmu_2 = dmu_1/dmu_2 dmu_2/dmu_3 ae (mu_3)`
• `mu_1 << mu_3`, `mu_2 << mu_3`, then `AA a,b in RR` `a mu_1 + beta mu_2 << mu_3` and `d(alpha mu_1 + beta mu_2)/dmu_3 = alpha dmu_1/dmu_3 + beta dmu_2/dmu_3 ae (mu_3)`
• `mu << nu` and `dmu/dnu > 0 ae (nu)` then `nu << mu` and `dnu/dmu = (dmu/dnu)^(-1)`
• Let `{mu_n}_(n>=1)` be a sequence of measures and `{a_n}_(n>=1) sub RR^+`, define `mu = sum a_n mu_n` then `mu << nu iff mu_n << nu AA n` and `dmu/dnu = sum a_n dmu_n/dnu ae (nu)`, `mu _|_ nu iff mu_n |_ n AA n >=1`

Signed measures

Let `mu_1, mu_2` be finite measures on MS `(Omega, F)`. Let `nu(A) = mu_1(A) - mu_2(A) AA A in F`. A finite signed measure satisfies:

• `nu(o\) = 0`
• `v(A) = sum v(A_i)`, `A_i` a partition of `A`
• `||nu|| = spr_"all partitions" {sum |v(A_i)|} < oo`.

Let `nu` be a finite signed measure and `|nu|(A) = spr_"all partitions" {sum |v(A_i)|}`, then `|nu|` is a finite measure.

A set function `nu` is a finite signed measure iff `EE mu_1, mu_2 "st" nu = mu_1 - mu_2` OR `EE mu` and `f in L^1(mu)` st `AA A in F v(A) = int_A f dmu`.

`A in F` is called a *negative set for `nu` if for any `B sub A` `nu(B) <= 0`.

Hahn decomposition theorem: `nu` a finite signed measure, then `EE` a positive set `Omega^+` and negative set `Omega^-` st `Omega = Omega^+ uu Omega^-` and `Omega^+ nn Omega^- = O/`. Jordan decomposition theorem: `nu = nu^+ - nu^_`.

`S = {nu: nu "is a finite signed measure"}` is a vector space, with total variation norm `||nu|| = |nu|(Omega)`.

Functions of bounded variation

`f: [a,b] -> RR`, then for some partition `Q`:

• positive variation of f `P(f, Q) = sum (f(x_i) - f(x_(i+1))_+`
• negative variation of f `N(f, Q) = sum (f(x_i) - f(x_(i+1))_-`
• total variation of f `P(f, Q) = sum |f(x_i) - f(x_(i+1)|`

And for without partition take supremum over all possible partitions. `f` is said to be of bounded variation if `T(f, [a,b]) < oo`. If `f in BV[a,b]` and `f_1 = N(f, [a,b]), f_2 = P(f, [a,b])` then `f_1, f_2` are non-decreasing and `f(x) = f_1(x) - f_2(x)`. `f in BV[a,b]` iff `EE` a finite signed measure `mu` on `(RR, B(RR)` st `f(x) = mu[a,x]`.

Absolutely continuous functions

A function is absolutely continuous if `AA epsi > 0 EE delta > 0 "st" I_j[a_j,b_j] "disjoint" and summ (b_j - a_j) < delta => sum |F(b_j) - F(a_j)| < epsi`. By mean value theorem, F differentiable and `F'` bounded, then F is ac.

A function `F:[a,b]->R` is absolutely continuous if `barF = F(a) I(x < a) + F(x) I(a<=x lt b) + F(b)I(x>=b)` is ac.

Fundamental theorem of Lebesgue integral calculus: `F: [a,b] -> RR` is ac iff there is a function `f: [a,b] -> RR` st `f` is Lebesgue measurable and integrable and `F(x) = F(a) + int_[[a,x]] f d_(mu_L) AA a<= x <= b`.