If A 0 is a null set

 
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dual
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dual
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The questioner has probably not yet seen the following answers.
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Guest123
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Hello Buri, sorry for digging out such an old thread, but I am currently working on this topic myself and therefore wanted to ask a little more detail. \ quoteon Hi question mark, under no circumstances can this definition 4.5.1 (from which book, must be clear, but I don't know, does not matter) be accepted. The principle must be that every subset of a zero set is also a zero set, otherwise one does not do justice to Lebesgue's actual concept, at least that seems to me that way. Greetings Buri \ quoteoff In my lecture on measure theory, and also in most of the books on measure theory that I work with, the zero set is defined as in the cited definition 4.5.1, i.e. that the zero set must be measurable and measure zero. Now I don't understand why one should demand that every subset of a null set must also be a null set again? In an incomplete measurement space, for example that of Borel Algebra, there are subsets of zero sets that cannot be measured and are therefore not zero sets. In complete measurement spaces, such as that of Lebesgue algebra, all subsets of zero sets are again contained in sigma algebra, i.e. measurable and thus zero sets. This means that a complete measurement space is simply defined by the fact that all subsets of zero sets are zero sets again. So why would you reject Definition 4.5.1? \ quoteon But with Definition 4.5.1 one turns in a circle: In order to define a zero set, the concept of the (complete!) measurement space is assumed, to define a (complete) measurement space, one needs the concept of the zero set. \ quoteoff Why do you need the concept of a complete measure space for the definition of a null set? If one does not demand at all (as in Definition 4.5.1, for example) that subsets of zero sets must necessarily be zero sets again, then one does not need a complete measurement space for the definition of a zero set. I would just have liked to hear your opinion again because, as I said, in the literature with which I am familiar it was always defined as in Def. 4.5.1.


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Buri
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Hi, what is definition 4.5.1 or where is a link to it? Greetings Buri


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Guest123
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Hello Buri, thanks for the answer. I meant the definition 4.5.1 that posted question marks: \ quoteon \ quoteoff


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zippy
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\ quoteon (2005-05-08 02:46 - Buri in article No. 7) under no circumstances can this definition 4.5.1 (from which book, must be clear, but I don't know, does not matter) be accepted. The principle must be that every subset of a null set is also a null set \ quoteoff This objection is simply wrong. A subset of a zero set does not have to be measurable in a (incomplete) measurement space. And so such a subset is not a zero set either. \ quoteon (2005-05-08 12:08 - Buri in article No. 11) But with definition 4.5.1 one turns in a circle: In order to define a zero set, the concept of the (complete!) measurement space is assumed, In order to define a (complete) measurement space, one needs the concept of the zero set. \ quoteoff This is also not correct: A zero set is defined as a measurable set with measure 0. Completeness is irrelevant with this definition.


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Guest123
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Alright, thank you very much. Then it's just as I thought it would be.


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