Measurability of the infimum of a class of functions


[1] Supremum of a class of functions

In the earlier post [1], we discussed measurability of the infimum of a class of measurable functions. In particular, for the infimum of a class of measurable functions as a function, we can show that it may not be measurable. Therefore, we shall need additional conditions to have the infimum function measurable. In this post, we show that the infimum function is

  • measurable if the class size is countable;
  • lower semicontinuous (thus measurable) if each function in the class is lower semicontinuous.

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Two representations of the fractional Laplacian operator

There are a few equivalent definitions for the fractional Laplacian operator {(-\Delta)^{\alpha}} for a constant {\alpha \in (0, 2)}. One is given by

\displaystyle - (- \Delta)^{\alpha} \phi(x) = C' \int_{\mathbb R^{d}} (\phi(x+y) - \phi(x) - D \phi(x) \cdot y I_{B_{1}}(y)) \nu(y) dy,

and the other one is given by

\displaystyle - (- \Delta)^{\alpha} \phi(x) = C \int_{\mathbb R^{d}} (\phi(x+y) + \phi(x-y) - 2\phi(x)) \nu(y) dy,

where {C} and {C'} are normalizing constants and {\nu(y) = \frac{1}{|y|^{d + \alpha}}} is a symmetric Levy measure on {\mathbb R^{d}}. In this below, we will show they are actually given equivalent with {2 C = C'}. Continue reading

Skorhod metric on a finite time interval

The notion of Skorohod metric provides a very useful topology in a discontinuous curve spaces, which is often arising as a sample space in stochastic analysis. Understanding Skorohod metric is also an interesting process for an undergraduate student, as a concrete example of a metric space but not a normed space. This note below is written by an undergraduate student with plenty of original examples.

PDF: here

Reference: [Bil99] ([Patrick Billingsley 1999])