Product measures
Product spaces and product measures
Given two measure spaces `(Omega_i, F_i, mu_i)` is it possible to construct a measure `mu_1 xx mu_2` on the product space `Omega_1 xx Omega_2` st `mu(A xx B) = mu(A) * mu(B)` for `A in F_1, B in F_2`?
- `Omega_1 xx Omega_2` is the Cartesian product of `Omega_1` and `Omega_2`
- the set `A_1 xx A_2`, `A_1 in F_1, A_2 in F_2`, is called a measurable rectangle, `C = ` all measurable rectangles
- the product `sigma`-algebra is `F_1 xx F_2 = sigma (: {A_1 xx A_2: A_1 in F_1, A_2 in F_2} :) = sigma(: C :)`
Starting with `mu` defined on `C`, `mu(A_1 xx A_2) = mu_1(A_1) * mu_2(A_2)`, we can use the extension procedure to extend `mu` to all `F_1 xx F_2`. Another approach, which allows us to calculate the values directly, proceeds as follows.
- `A_(omega_1) -= {omega_2 in Omega_2 : (omega_1, omega_2) in A}`, called the `omega_1`-section of A and is in `F_2`
- If `f: (Omega_1 xx Omega_2) -> Omega_3` is a `(: F_1 xx F_2, F_3 :)` measurable function, then the `omega_1`-section of `f` is `f_(omega_1)(omega_2): Omega_2 -> Omega_3 = f(omega_1, omega_2), omega_2 in Omega_2` and is `(: F_2, F_3 :)`
- `mu_1(A_(omega_2)) and mu_2(A_(omega_1))` are `F_1` and `F_2` measurable
- `mu_12(A) = int_(Omega_1) mu_2(A_(omega_1)) mu_1(d omega_1)` and `mu_21(A)` are measures on `F_1 xx F_2` and `mu_21(E) = mu_12(E) AA E in F_1 xx F_2`
- `mu_12 = mu_21 -= mu` is `sigma`-finite and is the only measure satisfying `mu(A_1 xx A_2) = mu_1(A_1)*mu_2(A_2) AA A_1 xx A_2 in C`
- the unique measure on `F_1 xx F_2` is called the product measure and is denoted `mu_1 xx mu_2`
- the measure space `(Omega_1 xx Omega_2, F_1 xx F_2, mu_1 xx mu_2)` is called the product measure space
Product space may not be complete, even if both original measures spaces are complete.
Fubini-Tonelli Theorems
Integral over product space can be treated as iterated integral if `f: Omega_1 xx Omega_2 -> R^+` (Tonelli’s therem) or `f in L_1(mu)` (Fubini’s theorem)
Tonelli: Let `(Omega_i, F_i, mu_i)` be `sigma`-finite spaces, `f` non-negative. Then:
- `g_1(omega_1) = int_(Omega_2) f(omega_1, omega_2) mu_2(d omega_2) : Omega_1 -> R^+` is `(: F_1, B(bar RR) :)` measurable
- `int_(Omega_1 xx Omega_2) f dmu = int_(Omega_1) g_1 dmu_1 = int_(Omega_2) g_2 dmu_2`
Fubini: If `f in L^1(mu)` then there exist sets `B_i in F_i` st
- `mu_i(B_i^c) = 0` for `i = 1,2`
- for `omega_1 in B_1, f(omega_1, *) in L_1(Omega_2, F_2, mu_2)`
- `g_1(omega_1) = int_(Omega_2) f(omega_1, omega_2) I_(B_1)(w_1) mu_2(d omega_2)` is measurable and `int_(Omega_1) g_1 dmu_1 = int_(Omega_1 xx Omega_2) f d(mu_1 xx dmu_2)`
Integration by parts: Let `F_1, F_2` be `uarr`, right-continuous functions on `[a,b]` with no common points of discontinuity. Then `int_((a,b]) F_1(x) dF_2(x)` `= F_1(b)F_2(b) - F_1(a)F_2(b) - int_((a,b]) F_2(x) dF_1(x)`. If `F_1, F_2` are ac with non-negative densities `f_1, f_2` then `int_a^b F_1(x) f_2(x) dx = F_1(b)F_2(b) - F_1(b)F_2(a) - int_a^b F_2(x) f_1(x) dx`. (Can always decompose into two non-negative functions so this also holds for all Lebesgue intergrable functions).
Extension to products of higher order use extension procedure.
Convolutions
Sequences
Given `{a_n}, {b_n} in L^1`, let `c_n = a ** b = sum_0^n a_j b_(n-j)`, then:
- `{c_n} in L^1`
- `sum c_n = (sum a_n)(sum b_n)`
- `C(s) = sum c_n s^n = A(s) * B(s)`
Functions
`f, g in L^1` then:- `h_1(x,y) = f(x-y) g(y)`, `h_2(x,y) = f(x) g(y-x)` are both `RR^2 -> RR`, Borel measurable
- `EE B in B_1(RR) "st" mu_2(B_1^c) = 0, x in B_1 => h_1(x_1, *) in L^1(mu_L)`
- `psi(x) = (int h_1(x,y) dmu_L(dy)) I_(B_1) = (psi(x)) = (f ** g)(x) in L^1(mu)`, and `int psi(x) dmu_L = (int f dmu_L)*(int f dmu_L)`