## The Signed Measure Monad

### Hardy Hulley – 1 December 2021

In this talk I will describe an adjunction between the category of measurable spaces and the category of AL spaces with linear contractions, which generalises the adjunction between the category of sets and the category of Banach spaces with linear contractions, due to Pumplun and Rorhl (1984). The description of the counit for this adjunction requires an appropriate theory of vector-valued integration. I will demonstrate that the Pettis integral does the job. The monad generated by the adjunction sends a measurable space to the space of signed measures defined on it, with the monad unit sending each element of a measurable space to the corresponding Dirac measure on that space and the monad multiplication sending each signed measure on the space of signed measures on a measurable space to its barycenter.

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