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SOLUTION OF EQUATION AX XB = C BY INVERSION OF AN M × M OR N ×N MATRIX ∗ ANTONY JAMESON† SIAM J Appl Math Vol 16, No 5, September 1968 It is often of interest to solve the equation. É C x g j ð Ý ¯ é C D ê ½ æ è g Ý ð µ Ä ¢ é B ¯ C ³ ç É £ § ð â µ I ð Å « é C D ê ½ æ è g Ý ð µ Ä ¢ é B a F ^ I Í ­ È ­ Æ à v C o V Û ì p Æ C ô ç p Ì Q Í p Ó ³ ê. á ¶ Å Ä · Ñ Ñ · Ý ¹ Ò Ø Ù Í Ï µ ¹ · Ò ¹ Ó Ó ´ Ô ¹ µ Ç ¼ ¹ Í ¸ Ç 5 h = > 4 0 9 2, 1?.

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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. T \ _ q , è a Ä ö cps50/51/52/54/56/58 ´  > & q 2 9 Í n u  n i n ' è é n. Lemma 112 If ¿ is a topology on X and B µ ¿;.

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