By Shlens J.
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Additional resources for A tutorial on Principal Component Analysis
A measurable set B is said to be negligible if μ(B) = 0. An arbitrary subset of E is said to be negligible if it is contained in a measurable negligible set. The measure space is said to be complete if every negligible set is measurable. If it is not complete, the following shows how to enlarge E to include all negligible sets and to extend μ onto the ¯ μ enlarged E. 16. The measure space (E, E, ¯) described is called the completion of (E, E, μ). When E = R and E = BR and ¯ are called the Lebesgue measurable sets.
Otherwise, if μ(f + ) = μ(f − ) = +∞, then μf is undeﬁned. 4 Remarks. Let f, g, etc. be simple and positive. a) The formula for μf remains the same even when f = ai 1Ai is not the canonical representation of f . This is easy to check using the ﬁnite additivity of μ. b) If a and b are in R+ , then af + bg is simple and positive, and the linearity property holds: μ(af + bg) = a μf + b μg. This can be checked using the preceding remark. c) If f ≤ g then μf ≤ μg. This follows from the linearity property above applied to the simple positive functions f and g − f : μf ≤ μf + μ(g − f ) = μ(f + g − f ) = μg.
The limitation to positive E-measurable functions can be removed: for arbitrary f in E the same formula holds provided that the integral on one side be well-deﬁned (and then both sides are well-deﬁned). The preceding theorem is a generalization of the change of variable formula from calculus. 3 ν(dx)f (h(x)) = μ(dy)f (y), F E that is, if h(x) is replaced with y then ν(dx) must be replaced with μ(dy). In calculus, it is often the case that E = F = Rd for some ﬁxed dimension d, and μ and ν are expressed in terms of the Lebesgue measure on Rd and the Jacobian of the transformation h.
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