The Unapologetic Mathematician

Mathematics for the interested outsider

Inner Measures

A quick one for today.

In analogy with the outer measure \mu^* induced on the hereditary \sigma-ring \mathcal{H}(\mathcal{S}) by the measure \mu on the \sigma-ring \mathcal{S}, we now define the “inner measure” \mu_* that \mu induces on the same hereditary \sigma-ring \mathcal{H}(\mathcal{S}). We’ve seen that the outer measure is

\displaystyle\mu^*(E)=\inf\{\mu(F)\vert E\subseteq F\mathcal{S}\}

Accordingly, the inner measure is a set function defined by

\displaystyle\mu_*(E)=\sup\{\mu(F)\vert E\supseteq F\mathcal{S}\}

In a way, the properties of \mu_* are “dual” to those of \mu^*. The easy ones are the same: it’s non-negative, monotone, and \mu_*(0)=0.

We could also define \mu_* in terms of the completed measure. Since \mathcal{S}\subseteq\overline{\mathcal{S}}, it’s clear

\displaystyle\mu_*(E)=\sup\{\mu(F)\vert E\supseteq F\mathcal{S}\}\leq\sup\{\bar{\mu}(F)\vert E\supseteq F\overline{\mathcal{S}}\}

On the other hand, the definition of the completion says that for every F\in\overline{\mathcal{S}} there is a G\in\mathcal{S} with G\subseteq F and \mu(G)=\bar{\mu}(F), and so this is actually an equality.

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April 8, 2010 - Posted by | Analysis, Measure Theory

3 Comments »

  1. [...] and measurable kernels. These will allow us to move statements we want to show about outer and inner measures to the realm of proper measures, where we can use nice things like [...]

    Pingback by Using Measurable Covers and Kernels I « The Unapologetic Mathematician | April 12, 2010 | Reply

  2. [...] a subset , we write for the image of under the transformation — . I say that the outer and inner Lebesgue measures are both nicely behaved under the transformation [...]

    Pingback by Lebesgue Measure and Affine Transformations « The Unapologetic Mathematician | April 22, 2010 | Reply

  3. [...] a measure space , we will routinely use without comment the associated outer measure and inner measure on the hereditary -ring [...]

    Pingback by Measurable Spaces, Measure Spaces, and Measurable Functions « The Unapologetic Mathematician | April 26, 2010 | Reply


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