Let ν be a signed measure on ( X , M ). We call the set E positive if ν ( F ) ≥ 0 for any F ⊆ E. For a sequence { E j } j = 1 ∞ of positive sets, it's easy to check that ⋃ 1 ∞ E j is also a positive set. Intuitively, the sum of positive measure is still positive. However, given a sequence { G i } i ∈ I of positive sets for uncountable index set I, do we have ⋃ i ∈ I G i being a positive set?Technically, I think it's not a positive set since the uncountable union of measurable set might not be in M anymore. Is my reasoning correct? However, I feel that the intuition "sum of positive measures should be positive" still holds here.

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Let   ν be a signed measure on   (  X  ,      M    ). We call the set   E positive if   ν  (  F  )  ≥  0 for any   F  ⊆  E. For a sequence   {      E    j        }          j      =      1              ∞       of positive sets, it&#039;s easy to check that       ⋃          1              ∞            E    j   is also a positive set. Intuitively, the sum of positive measure is still positive. However, given a sequence   {      G    i        }          i      ∈      I       of positive sets for uncountable index set I, do we have       ⋃          i      ∈      I            G    i   being a positive set?Technically, I think it&#039;s not a positive set since the uncountable union of measurable set might not be in       M   anymore. Is my reasoning correct? However, I feel that the intuition &quot;sum of positive measures should be positive&quot; still holds here.

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Consider the signed measure   ν  (  E  )  =      ∫          E        f  d  μ where   μ is Lebesgue measure. Note that   E is a positive set if   f  ≥  0 almost everywhere. However, the uncountable union accumulates null sets into a set with a non-zero measure. That is, let   f  =  −  1 and   E  =  (  0  ,  1  ]. Let   {      x    i        }                  x        i            ∈      E       be a sequence of positive sets. However,   ν  (      ⋃          i                  x      i        )  =  ν  (  E  )  =  −  1 which is not a positive set.

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