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Here is an elementary example from measure theory.

Usually L_p-spaces of a measurable space Z are defined with respect to some faithful measure μ on Z. More precisely, if p is a complex number with a nonnegative real part, then by definition L_p(Z,μ) consists of all functions f on Z such that μ(|f|^1/ℜp) is finite if ℜp>0. If ℜp=0, then L_p(Z,μ) consists of all bounded functions on Z. (Here I use the algebraic convention for L_p-spaces, namely L_p=L^1/p, in particular L₀=L^∞.)

Even though p is assumed to be complex, L_p(Z,μ) only depends on the real part of p. This will be fixed later.

It turns out that all spaces L_p(Z,μ) for different choices of a faithful measure μ are canonically isomorphic to each other. Suppose μ and ν are two faithful measures on Z and (Dμ:Dν) is the Radon-Nikodym derivative of μ with respect to ν, i.e., μ=(Dμ:Dν)ν. Then the map f∈L_p(Z,μ)⁠↦f(Dμ:Dν)^p∈L_p(Z,ν) is an isomorphism. Observe that these isomorphisms are compatible with each other (i.e., passing from λ to μ and then from μ to ν is the same as passing from λ to ν and passing from μ to itself is identity). Hence we have a compatible system of isomorphisms as described in Marty's answer, therefore we can denote its limit (or colimit) by L_p(Z). Thus we no longer need to choose a measure to define L_p-spaces.

Individual spaces L_p(Z,μ) depend only on the real part of p, but the isomorphisms between them also depend on the imaginary part of p. Therefore, if p−q is imaginary, then L_p(Z) is isomorphic to L_q(Z) non-canonically. There is no canonical isomorphism because such a canonical isomorphism would give us a canonical measure on Z.

Thus we got rid of the dependence on the choice of a measure and obtained a meaningful definition of L_p-space for compex values of p.

The spaces L₀(Z) and L₁(Z) can be defined canonically without this procedure: L₀(Z) is the space of all bounded functions on Z and L₁(Z) is the space of all finite complex-valued measures on Z. However, all constructions of L_p(Z) for p∉⁠{0,1} known to me involve some kind of limit/colimit over all measures.

Here is an elementary example from measure theory.

Usually L_p-spaces of a measurable space Z are defined with respect to some faithful measure μ on Z. More precisely, if p is a complex number with a nonnegative real part, then by definition L_p(Z,μ) consists of all functions f on Z such that μ(|f|^1/ℜp) is finite if ℜp>0. If ℜp=0, then L_p(Z,μ) consists of all bounded functions on Z. (Here I use the algebraic convention for L_p-spaces, namely L_p=L^1/p, in particular L₀=L^∞.)

Even though p is assumed to be complex, L_p(Z,μ) only depends on the real part of p. This will be fixed later.

It turns out that all spaces L_p(Z,μ) for different choices of a faithful measure μ are canonically isomorphic to each other. Suppose μ and ν are two faithful measures on Z and (Dμ:Dν) is the Radon-Nikodym derivative of μ with respect to ν, i.e., μ=(Dμ:Dν)ν. Then the map f∈L_p(Z,μ)⁠↦f(Dμ:Dν)^p∈L_p(Z,ν) is an isomorphism. Observe that these isomorphisms are compatible with each other (i.e., passing from λ to μ and then from μ to ν is the same as passing from λ to ν and passing from μ to itself is identity). Hence we have a compatible system of isomorphisms as described in Marty's answer, therefore we can denote its limit (or colimit) by L_p(Z). Thus we no longer need to choose a measure to define L_p-spaces.

Individual spaces L_p(Z,μ) depend only on the real part of p, but the isomorphisms between them also depend on the imaginary part of p. Therefore, if p−q is imaginary, then L_p(Z) is isomorphic to L_q(Z) non-canonically. There is no canonical isomorphism because such a canonical isomorphism would give us a canonical measure on Z.

Thus we got rid of the dependence on the choice of a measure and obtained a meaningful definition of L_p-space for complex values of p.

The spaces L₀(Z) and L₁(Z) can be defined canonically without this procedure: L₀(Z) is the space of all bounded functions on Z and L₁(Z) is the space of all finite complex-valued measures on Z. However, all constructions of L_p(Z) for p∉⁠{0,1} known to me involve some kind of limit/colimit over all measures.

Here is an elementary example from measure theory.

Usually L_p-spaces of a measurable space Z are defined with respect to some faithful measure μ on Z. More precisely, if p is a complex number with a nonnegative real part, then by definition L_p(Z,μ) consists of all functions f on Z such that μ(|f|^1/ℜp) is finite if ℜp>0. If ℜp=0, then L_p(Z,μ) consists of all bounded functions on Z. (Here I use the algebraic convention for L_p-spaces, namely L_p=L^1/p, in particular L₀=L^∞.)

Even though p is assumed to be complex, L_p(Z,μ) only depends on the real part of p. This will be fixed later.

It turns out that all spaces L_p(Z,μ) for different choices of a faithful measure μ are canonically isomorphic to each other. Suppose μ and ν are two faithful measures on Z and (Dμ:Dν) is the Radon-Nikodym derivative of μ with respect to ν, i.e., μ=(Dμ:Dν)ν. Then the map f∈L_p(Z,μ)⁠↦f(Dμ:Dν)^p∈L_p(Z,ν) is an isomorphism. Observe that these isomorphisms are compatible with each other (i.e., passing from λ to μ and then from μ to ν is the same as passing from λ to ν and passing from μ to itself is identity). Hence we have a compatible system of isomorphisms as described in Marty's answer, therefore we can denote its limit (or colimit) by L_p(Z). Thus we no longer need to choose a measure to define L_p-spaces.

Individual spaces L_p(Z,μ) depend only on the real part of p, but the isomorphisms between them also depend on the imaginary part of p. Therefore, if p−q is real, then L_p(Z) is isomorphic to L_q(Z) non-canonically. There is no canonical isomorphism because such a canonical isomorphism would give us a canonical measure on Z.

Thus we got rid of the dependence on the choice of a measure and obtained a meaningful definition of L_p-space for compex values of p.

The spaces L₀(Z) and L₁(Z) can be defined canonically without this procedure: L₀(Z) is the space of all bounded functions on Z and L₁(Z) is the space of all finite complex-valued measures on Z. However, all constructions of L_p(Z) for p∉⁠{0,1} known to me involve some kind of limit/colimit over all measures.

Here is an elementary example from measure theory.

Usually L_p-spaces of a measurable space Z are defined with respect to some faithful measure μ on Z. More precisely, if p is a complex number with a nonnegative real part, then by definition L_p(Z,μ) consists of all functions f on Z such that μ(|f|^1/ℜp) is finite if ℜp>0. If ℜp=0, then L_p(Z,μ) consists of all bounded functions on Z. (Here I use the algebraic convention for L_p-spaces, namely L_p=L^1/p, in particular L₀=L^∞.)

Even though p is assumed to be complex, L_p(Z,μ) only depends on the real part of p. This will be fixed later.

It turns out that all spaces L_p(Z,μ) for different choices of a faithful measure μ are canonically isomorphic to each other. Suppose μ and ν are two faithful measures on Z and (Dμ:Dν) is the Radon-Nikodym derivative of μ with respect to ν, i.e., μ=(Dμ:Dν)ν. Then the map f∈L_p(Z,μ)⁠↦f(Dμ:Dν)^p∈L_p(Z,ν) is an isomorphism. Observe that these isomorphisms are compatible with each other (i.e., passing from λ to μ and then from μ to ν is the same as passing from λ to ν and passing from μ to itself is identity). Hence we have a compatible system of isomorphisms as described in Marty's answer, therefore we can denote its limit (or colimit) by L_p(Z). Thus we no longer need to choose a measure to define L_p-spaces.

Individual spaces L_p(Z,μ) depend only on the real part of p, but the isomorphisms between them also depend on the imaginary part of p. Therefore, if p−q is imaginary, then L_p(Z) is isomorphic to L_q(Z) non-canonically. There is no canonical isomorphism because such a canonical isomorphism would give us a canonical measure on Z.

Thus we got rid of the dependence on the choice of a measure and obtained a meaningful definition of L_p-space for compex values of p.

The spaces L₀(Z) and L₁(Z) can be defined canonically without this procedure: L₀(Z) is the space of all bounded functions on Z and L₁(Z) is the space of all finite complex-valued measures on Z. However, all constructions of L_p(Z) for p∉⁠{0,1} known to me involve some kind of limit/colimit over all measures.

Here is an elementary example from measure theory.

Usually L_p-spaces of a measurable space Z are defined with respect to some faithful measure μ on Z. More precisely, if p is a complex number with a nonnegative real part, then by definition L_p(Z,μ) consists of all functions f on Z such that μ(|f|^1/ℜp) is finite if ℜp>0. If ℜp=0, then L_p(Z,μ) consists of all bounded functions on Z. (Here I use the algebraic convention for L_p-spaces, namely L_p=L^1/p, in particular L₀=L^∞.)

Even though p is assumed to be complex, L_p(Z,μ) only depends on the real part of p. This will be fixed later.

It turns out that all spaces L_p(Z,μ) for different choices of a faithful measure μ are canonically isomorphic to each other. Suppose μ and ν are two faithful measures on Z and (Dμ:Dν) is the Radon-Nikodym derivative of μ with respect to ν, i.e., μ=(Dμ:Dν)ν. Then the map f∈L_p(Z,μ)⁠↦f(Dμ:Dν)^p∈L_p(Z,ν) is an isomorphism. Observe that these isomorphisms are compatible with each other (i.e., passing from λ to μ and then from μ to ν is the same as passing from λ to ν and passing from μ to itself is identity). Hence we have a compatible system of isomorphisms as described in Marty's answer, therefore we can denote its limit (or colimit) by L_p(Z). Thus we no longer need to choose a measure to define L_p-spaces.

Individual spaces L_p(Z,μ) depend only on the real part of p, but the isomorphisms between them also depend on the imaginary part of p. Therefore, if p−q is real, then L_p(Z) is isomorphic to L_q(Z) non-canonically. There is no canonical isomorphism because such a canonical isomorphism would give us a canonical measure on Z.

Thus we got rid of the dependence on the choice of a measure and obtained a meaningful definition of L_p-space for compex values of p.

The spaces L₀(Z) and L₁(Z) can be defined canonically without this procedure: L₀(Z) is the space of all bounded functions on Z and L₁(Z) is the space of all finite complex-valued measures on Z. However, all constructions of L_p(Z) for p∉⁠{0,1} known to me involve some kind of limit/colimit over all measures.

Here is an elementary example from measure theory.

Usually L_p-spaces of a measurable space Z are defined with respect to some faithful measure μ on Z. More precisely, if p is a complex number with a nonnegative real part, then by definition L_p(Z,μ) consists of all functions f on Z such that μ(|f|^1/ℜp) is finite if ℜp>0. If ℜp=0, then L_p(Z,μ) consists of all bounded functions on Z. (Here I use the algebraic convention for L_p-spaces, namely L_p=L^1/p, in particular L₀=L^∞.)

Even though p is assumed to be complex, L_p(Z,μ) only depends on the real part of p. This will be fixed later.

It turns out that all spaces L_p(Z,μ) for different choices of a faithful measure μ are canonically isomorphic to each other. Suppose μ and ν are two faithful measures on Z and (Dμ:Dν) is the Radon-Nikodym derivative of μ with respect to ν, i.e., μ=(Dμ:Dν)ν. Then the map f∈L_p(Z,μ)⁠↦f(Dμ:Dν)^p∈L_p(Z,ν) is an isomorphism. Observe that these isomorphisms are compatible with each other (i.e., passing from λ to μ and then from μ to ν is the same as passing from λ to ν and passing from μ to itself is identity). Hence we have a compatible system of isomorphisms as described in Marty's answer, therefore we can denote its limit (or colimit) by L_p(Z). Thus we no longer need to choose a measure to define L_p-spaces.

Individual spaces L_p(Z,μ) depend only on the real part of p, but the isomorphisms between them also depend on the imaginary part of p. Therefore, if p−q is real, then L_p(Z) is isomorphic to L_q(Z) non-canonically. There is no canonical isomorphism because such a canonical isomorphism would give us a canonical measure on Z.

Thus we got rid of the dependence on the choice of a measure and obtained a meaningful definition of L_p-space for compex values of p.

The spaces L₀(Z) and L₁(Z) can be defined canonically without this procedure: L₀(Z) is the space of all bounded functions on Z and L₁(Z) is the space of all finite complex-valued measures on Z. However, all constructions of L_p(Z) for p∉⁠{0,1} known to me involve some kind of limit/colimit over all measures.