Let Q be an oriented smooth manifold. I am trying to construct a finite measure on Q, i.e. a top rank positive differential form with finite volume.My idea is given a collection of local charts ( U α , ψ α ), we can construct a positive μ α as μ α = f α d q 1 ∧ ⋯ ∧ d q n for any positive smooth f α on U α . Now using the paracompactness of any manifold, we can find a countable collection of regular coordinate balls that is an open locally finite refinement of U α . Say this collection is ( V n ). Then we can construct a global top rank form μ by a partition of unity subordinate to V n , μ = ∑ i ρ i μ i .I am trying to show that this has finite measure but I am struggling to finish the proof here. So we have ∫ Q μ = ∑ i ∫ V i ρ i f i d q 1 ∧ ⋯ ∧ d q n = ∑ i ∫ ψ i ( V i ) ( ρ i f i ∘ ψ i − 1 ) d n qwhere d n q is the Lebesgue measure.I think I need to use the compactness of the domains but I can't figure out how to show that this integral can be finite, or come up with a different construction. I would greatly appreciate any help.

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Let   Q be an oriented smooth manifold. I am trying to construct a finite measure on   Q, i.e. a top rank positive differential form with finite volume.My idea is given a collection of local charts   (      U    α    ,      ψ    α    ), we can construct a positive       μ    α   as       μ    α    =      f    α    d      q    1    ∧  ⋯  ∧  d      q    n   for any positive smooth       f    α   on       U    α  . Now using the paracompactness of any manifold, we can find a countable collection of regular coordinate balls that is an open locally finite refinement of       U    α  . Say this collection is (      V    n  ). Then we can construct a global top rank form   μ by a partition of unity subordinate to       V    n  ,  μ  =      ∑    i        ρ    i        μ    i    .I am trying to show that this has finite measure but I am struggling to finish the proof here. So we have       ∫    Q    μ  =      ∑    i        ∫                  V        i                  ρ    i        f    i    d      q    1    ∧  ⋯  ∧  d      q    n    =      ∑    i        ∫                  ψ        i            (              V        i            )        (      ρ    i        f    i    ∘      ψ    i          −      1        )      d    n    qwhere       d    n    q is the Lebesgue measure.I think I need to use the compactness of the domains but I can&#039;t figure out how to show that this integral can be finite, or come up with a different construction. I would greatly appreciate any help.

Eli Shaffer · Accepted Answer

There is no reason to expect   ∫  μ to be finite in general. For instance, imagine   Q  =      R   and every       μ    i   is just   d  x; then you&#039;d be integrating   d  x over all of       R   which diverges.The trick, though, is that there is no reason you need to use       ρ    i   themselves as the coefficients on the       μ    i  . All you care is that you pick some coefficients with compact support such that   μ still ends up being nonzero everywhere. So for instance, you could instead take   μ  =  ∑      c    i        ρ    i        μ    i   for any sequence of positive constants       c    i  . If you choose the       c    i   appropriately (e.g., choose       c    i   so that   ∫      c    i        ρ    i        μ    i    =  1      /        2    i  ), you can make   ∫  μ finite.

Let Q be an oriented smooth manifold. I am trying to construct a finite measure on Q , i.

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