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Probability Distributions CEE 1L Uncertainty, Design, and Optimization Department of Civil and Environmental Engineering Duke University Philip Scott Harvey, Henri P Gavin and Jeffrey T Scruggs.

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Remark 34 Alsonotethat, wehaveprovenin(e)ofTheorem32thatµ∗(A) ≤ µ∗(A T E) µ∗(A T EC) so that we need only to show the reverse inequality to establish that the set E is measurable The proof of the following elementary facts are left as exercise. @ a b 5 c 1 d ’!. Supreme_End_of_Education^ÂÎÕ^ÂÎÕBOOKMOBI Åâ `* / 6³ =« Dê KŠ Ræ ZI al gú n5 t¤ {ˆ ‚ ‰¡ ‘ ˜X"žó$¥„&¬ (²^*¹ù,ÀVÇW0ÌK2Ñ 4Õ‡6Úû8âRê ð >÷^@þ·B ÀD jF ƒH J ÊL (‰N /äP 7©R >pT F V MHX U(Z \¹\ cä^ k ` r b y!d €£f ‡”h ˜j “•l ™În Âp §Ÿr ®èt µ²v ¼ x Äz Ë/ ÒZ~ Ù^€ àö‚ 秄 ï^† õЈ ýÆŠ 6Œ MŽ Y.

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Design and development The CX was essentially an enlarged development of the Albatros CVII designed to take advantage of the new Mercedes DIVa engine that became available in 1917Unlike the CVII that preceded it in service, the CX utilised the top wing sparmounted radiator that had first been tried on the CV/17Other important modernisation features included provision for oxygen for. We have gOi = gHai. Proposition 4 All sets with µ∗(E) = 0 are µ∗ measurable Proof If µ∗(E) = 0, then for an arbitrary set A⊂Xwe have µ∗(A) ≥µ∗(A\E) = µ∗(A\E) µ∗(A∩E) {z } 0, because A∩E⊂E Let M∗ be the class of all µ∗measurable sets Theorem 5 (Carath´eodory) M∗ is a σalgebra and µ∗ M∗ →0,∞ is a measure Proof We will split the proof into several steps.

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