20 Background Group representation A group representation of Gin the vector space V is a function ρ : G →GL(V) (where GL(V) is the general linear group on V) such that for all g,g′ ∈Gρ(g ◦g′)=ρ(g) ◦ρ(g′) andρ(e)=IV, where IV is the identity matrix. Intuitively, a group representation is essentially a group action that acts onV linearly. Direct sum of representations The direct sumof two group representations ρ1 : G→GL(V) in V and ρ2 : G→GL(V′) in V′ is a group representation ρ1 ⊕ρ2 : G→GL(V ⊕V′) over the direct sumV ⊕V′, defined for v ∈V and v′ ∈V′ as (ρ1 ⊕ρ2)(g) · (v,v′)=(ρ1(g) · v,ρ2(g) · v′). (2.15) 2.4 Measure Theory Measure theory is a branch of mathematics that can serve as a generalisation and formalisation of (among others) probability theory. It provides a means to formulate metrics in the context of both VAEs, which are probabilistic models; and group theory, which is used to formulate symmetry-based disentanglement by Higgins et al. (2018). In this section, we summarise some relevant concepts from measure theory, which we use to formulate a quantification for symmetrybased disentanglement in Chapter 4. For a more thorough overview of measure theory, see e.g. the books by Fremlin (2000). σ-Algebra, measurable space, measurable set Aσ-algebra on a set Xis a collectionAof subsets of Xsuch that 1. ∅,X∈A, i.e. Acontains the empty set ∅ andXitself; 2. if A∈AthenAC ∈A, i.e. Ais closed under complements; 3. if Ai ∈Afor i ∈Nthen ∞[ i=1 Ai ∈A, i.e. Ais closed under countable unions.
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