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Measure Theory

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Measurable Spaces

Algebra

Let S be some abstract set, and let &Sigma._0 be a collection of subsets of S.
Then &Sigma._0 is an ~{algebra} if

  1. &empty. &in. &Sigma._0; _ S &in. &Sigma._0
  2. if A &in. &Sigma._0 then A^c &in. &Sigma._0
  3. if A_1, A_2 &in. &Sigma._0 then A_1 &intersect. A_2 &in. &Sigma._0 (~{finite intersection})

Sigma-algebra

A collection , &Sigma., of subsets of S is a &sigma.-~{algebra} if.

  1. &empty. &in. &Sigma._0; _ S &in. &Sigma._0
  2. if A &in. &Sigma._0 then A^c &in. &Sigma._0
  3. if A_{~i} &in. &Sigma., {~i} &in. &naturals., then &intersect._{~i} A_{~i} &in. &Sigma. (~{countable intersection})

Measurable Space

If S is an abstract set with an algebra &Sigma. defined on it, then the pair, (S, &Sigma.) is called a ~{measurable space}.

Sigma-algebra generated by subsets

S abstract set. @C &subseteq. 2^S (power set of S). We use the notation _ &sigma.(@C) _ to indicate the smallest &sigma.-algebra containing @C.
This always exists as 2^S is a &sigma.-algebra

&sigma.(@C) is called the &sigma.-algebra ~{generated} by @C

Borel Sigma-algebra

If S is a topological space then the ~{Borel} &sigma.-~{algebra} on S, @B(S), is the &sigma.-algebra generated by the collection of all open sets of S

Suppose that S = &reals. and let &pi. (&reals.) = \{ (&minus. &infty., ~x] : ~x &in. &reals. \} _ then _ @B (&reals.) = &sigma. (&pi. (&reals.))

[By convention, write @B for @B (&reals.)]

Set Functions

&Sigma._0 algebra on set S, a function &mu.: &Sigma. &rightarrow. [0, &infty.] is a ~{set function}.

Finite Additivity

A_1, A_2 &in. &Sigma._0, and A_1 &intersect. A_2 = &empty. _ then &mu. is ~{finitely additivity} if

&mu.(A_1 &union. A_2) _ = _ &mu.(A_1) + &mu.(A_2).

Countable Additivity

A_{~i} &in. &Sigma._0, {~i} &in. &naturals., and A_{~i} &intersect. A_{~j} = &empty. when ~i &neq. ~j, _ then &mu. is ~{countably additivity} if

&mu. rndb{ union{ A_{~i},{~i = 1},&infty.}} _ = _ sum{ &mu.(A_{~i}),{~i = 1},&infty.}.

Measure

If (S, &Sigma.) is a measurable space, &mu.: &Sigma. &rightarrow. [0, &infty.] is a ~{measure} if it is countably additive.

Measure spaces

If &mu. is a measure on (S, &Sigma.) the triple (S, &Sigma., &mu.) is called a ~{measure space} .

Pi-systems

@I is a &pi.-system if _ I_1, I_2 &in. @I _ &imply. _ I_1 &intersect. I_2 &in. @I.

Let &Sigma. = &sigma.(@I), and let &mu._1 and &mu._2 be measures on (S, &Sigma.) such that
_ &mu._1(S) = &mu._2(S) < &infty. _ and _ &mu._1 = &mu._2 on @I. _ Then _ &mu._1 = &mu._2

Caratheodory Extension

Set S, with algebra &Sigma._0 on S, and let &Sigma. = &sigma.(&Sigma._0)

If &mu._0 is a countably additive map, _ &mu._0: &Sigma._0 &rightarrow. [0, &infty.], _ then &exist. a measure &mu. on (S, &Sigma.) such that _ &mu. = &mu._0 _ on &Sigma._0

Lebesgue Measure

S = (0, 1], _ F &subseteq. S, _ F &in. &Sigma._0 _ if F can be written as finite union:

F _ = _ ( ~a_1, ~b_1] &union. ... &union. ( ~{a_r}, ~{b_r}] where ~r &in. &naturals., _ and _ 0 &le. ~a_1 &le. ~b_1 &le. ... &le. ~{a_r}&le. ~{b_r} &le. 1
Then &Sigma._0 is an algebra, _ put

&mu._0(F) _ = _ sum{(~{b_i} &minus. ~{a_i}), ~i = 1, ~r} then &mu._0 is well-defined and additive. In fact &mu._0 is countably additive.
Let &Sigma. = &sigma.(&Sigma._0), _ then &Sigma. = @B ( 0, 1] _ #{:}= @B( ( 0, 1] )
So by Caratheodory &exist. unique measure &mu. on ( (0, 1], @B(0, 1] ) _ which extends &mu._0 on &Sigma._0.
&mu. is called the ~{Lebesgue measure} on ( (0, 1], @B(0, 1] ) and is denoted Leb.
Similarly we can construct a Lebegue measure on ( &reals., @B ), which is also denoted Leb.

Measure Inequalities

Monotone-convergence of measures

Let ( S, &Sigma., &mu. ) be a measure space. F_{~n} &in. &Sigma.. We write F_{~n} &uparrow. F _ if _ F_{~n} &subseteq. F_{~n+1} (&forall. ~n &in. &naturals.) _ and _ &union. F_{~n} = F

  1. F_{~n} &in. &Sigma. ( ~n &in. &naturals.) and F_{~n} &uparrow. F _ then _ &mu.(F_{~n}) &uparrow. &mu.(F)
  2. G_{~n} &in. &Sigma. ( ~n &in. &naturals.), G_{~n} &downarrow. G and &mu.(G_{~k}) < &infty., some ~k _
    then _ &mu.(G_{~n}) &downarrow. &mu.(G)
  3. The union of a countable number of &mu.-null sets is &mu.-null.