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mathcamp02 - Math Camp 2: Probability Theory Sasha Rakhlin...

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Unformatted text preview: Math Camp 2: Probability Theory Sasha Rakhlin σ-algebra A σ-algebra Σ over a set Ω is a collection of subsets of Ω that is closed under countable set operations: 1. ∅ ∈ Σ. 2. E ∈ Σ then so is the complement of E . 3. If F is any countable collection of sets in Σ, then the union of all the sets E in F is also in Σ. Measure A measure µ is a function defined on a σ-algebra Σ over a set Ω with values in [0 , ∞ ] such that 1. The empty set has measure zero: µ ( ∅ ) = 2. Countable additivity: if E 1 , E 2 , E 3 , ... is a countable sequence of pairwise disjoint sets in Σ, µ ⎛ ⎝ ∞ [ E i ⎞ ⎠ ∞ = µ ( E i ) i =1 i =1 The triple (Ω , Σ , µ ) is called a measure space . Lebesgue measure The Lebesgue measure λ is the unique complete translation- invariant measure on a σ-algebra containing the intervals in IR such that λ ([0 , 1]) = 1. Probability measure Probability measure is a positive measure µ on the mea- surable space (Ω , Σ) such that µ (Ω) = 1.1....
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This note was uploaded on 11/11/2011 for the course BIO 9.07 taught by Professor Ruthrosenholtz during the Spring '04 term at MIT.

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mathcamp02 - Math Camp 2: Probability Theory Sasha Rakhlin...

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