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4). 8) g. 8) and A. 9) we can come back to M, on letting /2 = ... = In = 0. 26 IV. Axiomatic Theory of the Integral 3. Union and intersection of functions By the union of functions f and g whose values are in a given Banach space we mean the real valued function g= max

Then h~l+g and hence Sh~SI+Sg=O. 2. , In -+ I except for a null set X. , f = g except for a null set Y. Then fn -+ g outside XU Y. This g is another limit almost everywhere. Assume conversely that In -+ I except for X and In -+ g except for Y. Then 1= g except for XU Y. 3. , then Aln -+ AI lor any real number A. That is, the product by a number 01 a sequence convergent almost everywhere converges almost everywhere to the product 01 its limit by that number. Proof. It follows from In ~ gn -+ I that Aln ~ Agn -+ AI.

If f and g are measurable functions and one of them is real valued, then the function 1;1 g is measurable. Proof. Suppose first that g;;;. O. n g-+Iflf g. 2. Suppose now that g is real, but not necessarily non-negative. Then the assertion follows from the equality 1;1 g= 1;1 g+ -1;1 (- g-) and the preceding result. Finally, let f be real and g arbitrary. Then the assertion follows from the identity by what we have proved above. 9 (Fatou). If fn e U, fn -+ f as. and Ilfn los; M, then feU and Ilfl os; M.