152 L07 : Proof of Lemmas 4-6 for the outer-measure extension construction

Video 152.1: Lesson 7: Proof of Lemmas 4–6 for the outer-measure extension construction.

152.1 Lesson map

This lesson proves the next three lemmas from the extension roadmap.

• Recap: extending P from a field \mathcal{F}_0 to \mathcal{F}=\sigma(\mathcal{F}_0).
• Recall the outer measure P^*.
• Important warning: do not use that P^* is a probability measure yet.
• Recall the definition of \mathcal{M}.
• Lemma 4:
• Lemma 5:
• Lemma 6:

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Figure 152.1: And me with a longer caption that is more descriptive of the image, and can replace the image if it doesn’t load.

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Figure 152.2: And me with a longer caption that is more descriptive of the image, and can replace the image if it doesn’t load.

Figure 152.3

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Figure 152.5

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Figure 152.6: And me with a longer caption that is more descriptive of the image, and can replace the image if it doesn’t load.

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Figure 152.7: And me with a longer caption that is more descriptive of the image, and can replace the image if it doesn’t load.

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Figure 152.8: And me with a longer caption that is more descriptive of the image, and can replace the image if it doesn’t load.

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Figure 152.9: And me with a longer caption that is more descriptive of the image, and can replace the image if it doesn’t load.

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Figure 152.10: And me with a longer caption that is more descriptive of the image, and can replace the image if it doesn’t load.

outer measure definition.

Figure 152.11: And me with a longer caption that is more descriptive of the image, and can replace the image if it doesn’t load.

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Figure 152.12: And me with a longer caption that is more descriptive of the image, and can replace the image if it doesn’t load.

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Figure 152.13: And me with a longer caption that is more descriptive of the image, and can replace the image if it doesn’t load.

Figure 152.14