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| | As was noted, the normal form of an object $M\in\M$ is a "selected representative" from the equivalence class $[M]$, usually possessing some nice properties. The set of all these "representatives" intersect each equivalence class exactly once; such set is called a ''transversal'' (for the given equivalence relation). Existence of a transversal is ensured by the [[axiom of choice]] for arbitrary equivalence relation on arbitrary set. However, a transversal in general is far from being nice. For example, consider the equivalence relation "$x-y$ is rational" for real numbers $x,y$. Its transversal (so-called [[Non-measurable set|Vitali set]]) cannot be Lebesgue measurable! | | As was noted, the normal form of an object $M\in\M$ is a "selected representative" from the equivalence class $[M]$, usually possessing some nice properties. The set of all these "representatives" intersect each equivalence class exactly once; such set is called a ''transversal'' (for the given equivalence relation). Existence of a transversal is ensured by the [[axiom of choice]] for arbitrary equivalence relation on arbitrary set. However, a transversal in general is far from being nice. For example, consider the equivalence relation "$x-y$ is rational" for real numbers $x,y$. Its transversal (so-called [[Non-measurable set|Vitali set]]) cannot be Lebesgue measurable! |
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| − | Typically, the set $\M$, endowed with its natural σ-algebra, is a [[standard Borel space]], and the set $\{(x,y)\in\M\times\M:x\sim y\}$ is a Borel subset of $\M\times\M$; this case is well-known as a "Borel equivalence relation". | + | Typically, the set $\M$, endowed with its natural σ-algebra, is a [[standard Borel space]], and the set $\{(x,y)\in\M\times\M:x\sim y\}$ is a Borel subset of $\M\times\M$; this case is well-known as a "Borel equivalence relation". Still, existence of a ''Borel'' transversal is not guaranteed (for an example, use the Vitali set again). |
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| − | ---------------------------------------------
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| − | $\newcommand{\Om}{\Omega}
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| − | \newcommand{\A}{\mathcal A}
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| − | \newcommand{\B}{\mathcal B}
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| − | \newcommand{\M}{\mathcal M} $
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| − | A '''measure space''' is a triple $(X,\A,\mu)$ where $X$ is a set, $\A$ a [[Algebra of sets|σ-algebra]] of its subsets, and $\mu:\A\to[0,+\infty]$ a [[measure]]. Thus, a measure space consists of a [[measurable space]] and a measure. The notation $(X,\A,\mu)$ is often shortened to $(X,\mu)$ and one says that $\mu$ is a measure on $X$; sometimes the notation is shortened to $X$.
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| − | ====Basic notions and constructions====
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| − | ''Inner measure'' $\mu_*$ and ''outer measure'' $\mu^*$ are defined for all subsets $A\subset X$ by
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| − | : $ \mu_*(A) = \max\{\mu(B):B\in\A,B\subset A\}\,,\quad
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| − | \mu^*(A) = \min\{\mu(B):B\in\A,B\supset A\}\,;$
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| − | {{Anchor|null}}{{Anchor|full}}{{Anchor|almost}}
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| − | $A$ is called a ''null'' (or ''negligible'') set if $\mu^*(A)=0$; in this case the complement $X\setminus A$ is called a set of ''full measure'' (or ''conegligible''), and one says that $x\notin A$ for ''almost all'' $x$ (in other words, ''almost everywhere''). Two sets $A,B\subset X$ are ''almost equal'' (or ''equal mod 0'') if $(x\in A)\iff(x\in B)$ for almost all $x$ (in other words, $A\setminus B$ and $B\setminus A$ are negligible). Two functions $f,g:X\to Y$ are ''almost equal'' (or ''equal mod 0'', or ''equivalent'') if they are equal almost everywhere.
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| − | The ''[[Measure#product|product]]'' of two (or finitely many) measure spaces is a well-defined measure space.
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| − | A ''[[probability space]]'' is a measure space $(X,\A,\mu)$ satisfying $\mu(X)=1$. The product of infinitely many probability spaces is a well-defined probability space. (See {{Cite|D|Sect. 8.2}}, {{Cite|B|Sect. 3.5}}, {{Cite|P}}.)
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| − | ====Completion====
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| − | A subset $A\subset X$ is called ''measurable'' (or $\mu$-measurable) if it is almost equal to some $B\in\A$. In this case $\mu_*(A)=\mu^*(A)=\mu(B)$. If $\mu_*(A)=\mu^*(A)<\infty$ then $A$ is $\mu$-measurable. All $\mu$-measurable sets are a σ-algebra $\A_\mu$ containing $\A$.
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| − | Every $\A_\mu$-measurable function $X\to\R$ is almost equal to some $\A$-measurable function $X\to\R$. The same holds for arbitrary [[Measurable space#countably generated|countably generated measurable space]] in place of $\R$.
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| − | ''Example.'' Let $X$ be the real line, $\A$ the Borel σ-algebra and $\mu$ Lebesgue measure on $\A$, then $\A_\mu$ is the Lebesgue σ-algebra.
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| − | Let $(X,\A,\mu)$ be a measure space. Both $(X,\A,\mu)$ and $\mu$ are called ''complete'' if $\A_\mu=\A$ or, equivalently, if $\A$ contains all null sets. The ''completion'' of $(X,\A,\mu)$ is the complete measure space $(X,\A_\mu,\tilde\mu)$ where $\tilde\mu(A)=\mu(B)$ whenever $A\in\A_\mu$ is almost equal to $B\in\A$.
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| − | Let $(X,\A,\mu)$ be complete, and $\B\subset\A$ a sub-σ-algebra. Then $(X,\A,\mu)$ is the completion of $(X,\B,\mu|_\B)$ if and only if for every $A\in\A$ there exist $B,C\in\B$ such that $B\subset A\subset C$ and $\mu(C\setminus B)=0$.
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| − | Surprisingly, the Borel σ-algebra can be "almost restored" from the Lebesgue σ-algebra in the following sense.
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| − | Let $(X,\A,\mu)$ be complete, and $\B_1\subset\A$, $\B_2\subset\A$ two countably generated sub-σ-algebras such that $(X,\A,\mu)$ is both the completion of $(X,\B_1,\mu|_{\B_1})$ and the completion of $(X,\B_2,\mu|_{\B_2})$. Then there exists a set $Y\in\B_1\cap\B_2$ of full measure such that $\B_1|_Y=\B_2|_Y$. (Here $\B_i|_Y=\{B\cap Y:B\in\B_i\}=\{B\in\B_i:B\subset Y\}$.)
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| − | ====Isomorphism====
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| − | A ''strict isomorphism'' (or ''point isomorphism'', or ''[[metric isomorphism]]'') between two measure spaces $(X_1,\A_1,\mu_1)$ and $(X_2,\A_2,\mu_2)$ is a bijection $f:X_1\to X_2$ such that, first, the conditions $A_1\in\A_1$ and $A_2\in\A_2$ are equivalent whenever $A_1\subset X_1$, $A_2\subset X_2$, $A_2=f(A_1)$, and second, $\mu_1(A_1)=\mu_2(A_2)$ under these conditions.
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| − | A ''mod 0 isomorphism'' (or ''almost isomorphism'') between two measure spaces $(X_1,\A_1,\mu_1)$ and $(X_2,\A_2,\mu_2)$ is a strict isomorphism between some full measure sets $Y_1\in\A_1$ and $Y_2\in\A_2$ treated as measurable subspaces.
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| − | Thus we have two equivalence relations between measure spaces: ''"strictly isomorphic"'' and ''"almost isomorphic"''. (See {{Cite|I|Sect. 2.4}}, {{Cite|B|Sect. 9.2}}.)
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| − | If two measure spaces are almost isomorphic then clearly their completions are almost isomorphic. The converse, being wrong in general, surprisingly holds in the following important case.
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| − | Let measure spaces $(X_1,\A_1,\mu_1)$, $(X_2,\A_2,\mu_2)$ be such that (a) their completions are almost isomorphic, and (b) measurable spaces $(X_1,\A_1)$, $(X_2,\A_2)$ are countably generated. Then $(X_1,\A_1,\mu_1)$, $(X_2,\A_2,\mu_2)$ are almost isomorphic (under the same isomorphism, restricted to a smaller subset of full measure).
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| − | For ''complete'' measure spaces the two notions of isomorphism nearly coincide, as explained below.
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| − | An almost isomorphism between complete measure spaces $(X_1,\A_1,\mu_1)$, $(X_2,\A_2,\mu_2)$, being a bijection $Y_1\to Y_2$ between full measure sets $Y_1\subset X_1$, $Y_2\subset X_2$, extends readily to a strict isomorphism $X_1\to X_2$, since ''all'' maps are measurable on negligible sets $X_1\setminus Y_1$, $X_2\setminus Y_2$. The only possible obstacle is, different cardinalities of these negligible sets. The conclusion follows.
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| − | Assume that $(X_1,\A_1,\mu_1)$ is a complete measure space, $X_1$ is of cardinality continuum and contains some negligible set of cardinality continuum. Assume that $(X_2,\A_2,\mu_2)$ satisfies the same conditions. If $(X_1,\A_1,\mu_1)$, $(X_2,\A_2,\mu_2)$are almost isomorphic then they are strictly isomorphic.
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| − | Cardinality continuum is typical, but the fact holds in general, under the following condition: for every negligible set in every one of the two measure spaces there exists a negligible set of the same cardinality in the other measure space. (This argument is used, somewhat implicitly, in {{Cite|F|Vol. 3, Sect. 344I}}.)
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| − | ====Finite and σ-finite====
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| − | Let $(X,\A,\mu)$ be a measure space. Both $(X,\A,\mu)$ and $\mu$ are called ''totally finite'' if $\mu(X)<\infty$, and ''σ-finite'' if $X$ can be split into countably many sets of finite measure, that is, $X=A_1\cup A_2\cup\dots$ for some $A_n\in\A$ such that $\forall n \;\; \mu(A_n)<\infty$. (Totally finite measures are also σ-finite.)
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| − | ====Perfect and standard====
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| − | Let $(X,\A,\mu)$ be a totally finite measure space. Both $(X,\A,\mu)$ and $\mu$ are called [[Perfect measure|''perfect'']] if for every $\mu$-measurable (or equivalently, for every $\A$-measurable) function $f:X\to\R$ the image $f(X)$ contains a Borel (or equivalently, σ-compact) subset $B$ whose preimage $f^{-1}(B)$ is of full measure. (See {{Cite|B|Sect. 7.5}}.)
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| − | For ''[[standard probability space]]s'' see the separate article. Standard measure spaces are defined similarly. They are perfect, and admit a complete classification.
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| − | ''Examples.'' The real line with Lebesgue measure on Borel σ-algebra is an incomplete σ-finite measure space. The real line with Lebesgue measure on Lebesgue σ-algebra is a complete σ-finite measure space. The unit interval $(0,1)$ with Lebesgue measure on Lebesgue σ-algebra is a standard probability space. The product of countably many copies of this space is standard; for uncountably many factors the product is perfect but nonstandard. The one-dimensional [[Hausdorff measure]] on the plane is not σ-finite.
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| − | ====Atoms and continuity====
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| − | Let $\mu(X)<\infty$. An ''atom'' of $(X,\A,\mu)$ (and of $\mu$) is a non-negligible measurable set $A\subset X$ such that every measurable subset of $A$ is either negligible or almost equal to $A$. Both $(X,\A,\mu)$ and $\mu$ are called ''atomless'' or ''nonatomic'' (or ''diffused'') if they have no atoms; on the other hand, they are called ''purely atomic'' if there exists a partition of $X$ into atoms. (See {{Cite|D|Sect. 3.5}}, {{Cite|B|Sect. 1.12(iii)}}, {{Cite|M|Sect. 6.4.1}}.)
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| − | If $x\in X$ is such that the single-point set $\{x\}$ is a non-negligible measurable set then clearly $\{x\}$ is an atom. If $(X,\A,\mu)$ is standard then every atom is almost equal to some $\{x\}$, but in general it is not.
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| − | Let $\{x\}$ be measurable for all $x\in X$. Both $(X,\A,\mu)$ and $\mu$ are called ''continuous'' if $\mu(\{x\})=0$ for all $x\in X$; on the other hand, they are called ''discrete'' if $X$ is almost equal to some finite or countable set. (See {{Cite|C|Sect. 1.2}}, {{Cite|K|Sect. 17.A}}.) A discrete space cannot be atomless (unless $\mu(X)=0$), but a purely atomic nonstandard space can be continuous. (See {{Cite|B|Sect. 7.14(v)}}.)
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| − | See also "taxonomy of measure spaces" in {{Cite|F|Vol. 2, Chapter 21}}.
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| − | ====Sub-σ-algebras and linear sublattices====
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| − | Throughout this section we restrict ourselves to σ-algebras that contain all null sets. That is, a measure space $(X,\A,\mu)$ is assumed to be complete, and whenever we call $\B\subset\A$ a sub-σ-algebra we also assume that $\B$ contains all null sets: $\forall A\in\A \; (\, \mu(A)=0 \,\Longrightarrow A\in\B \,)$. This assumption is stronger than completeness of the measure space $(X,\B,\mu|_\B)$.
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| − | Every sub-σ-algebra $\B\subset\A$ leads to a subspace (that is, closed linear subset) $L_2(X,\B,\mu|_\B)$ of the Hilbert space $L_2(X,\A,\mu)$.
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| − | '''Theorem''' (Šidák). A subspace $H\subset L_2(X,\A,\mu)$ is of the form $L_2(X,\B,\mu|_\B)$ for some sub-σ-algebra $\B\subset\A$ if and only if $\bsone\in H$ and $\forall f\in H \; (\, |f|\in H \,)$. (Here $\bsone(x)=1$ and $|f|(x)=|f(x)|$ for $x\in X$.)
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| − | Due to linearity, the condition $\forall f\in H \; (\, |f|\in H \,)$ is equivalent to the condition $\forall f,g\in H \; (\, f\land g\in H \text{ and } f\lor g\in H \,)$ where $(f\land g)(x)=\min(f(x),g(x))$ and $(f\lor g)(x)=\max(f(x),g(x))$ for $x\in X$; that is, $H$ is a linear sublattice of the linear lattice $L_2(X,\A,\mu)$.
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| − | ====On terminology====
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| − | The word "isomorphic" (for measure spaces) is interpreted as ''almost isomorphic'' in {{Cite|I|Sect. 2.4}} (which is usual according to {{Cite|B|Sect. 9.2}}) but as ''strictly isomorphic'' in {{Cite|F|Vol. 2, Sect. 254, Notes and comments}}; there, the notion ''almost isomorphic'' is only mentioned in passing as "nearly an isomorphism".
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| − | The phrase "separable measure space" is quite ambiguous. Some authors call $(X,\A,\mu)$ separable when the Hilbert space $L_2(X,\A,\mu)$ is separable; equivalently, when $\A$ contains a countably generated sub-σ-algebra $\B$ such that every set of $\A$ is almost equal to some set of $\B$. (See {{Cite|B|Sect. 7.14(iv)}}, {{Cite|M|Sect. IV.6.0}}.) But in {{Cite|I|Sect. 3.1}} it is required instead that $\B$ separates points and $(X,\A,\mu)$ is complete, while in {{Cite|H}} all these conditions are imposed together.
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| − | The phrase "measurable space" is avoided in {{Cite|F}} "as in fact many of the most interesting examples of such objects have no useful measures associated with them" {{Cite|F|Vol. 1, Sect. 111B}}.
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| − | ====References====
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| − | {|
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| − | |valign="top"|{{Ref|T}}|| Terence Tao, "An introduction to measure theory", AMS (2011). {{MR|2827917}} {{ZBL|05952932}}
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| − | |-
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| − | |valign="top"|{{Ref|C}}|| Donald L. Cohn, "Measure theory", Birkhäuser (1993). {{MR|1454121}} {{ZBL|0860.28001}}
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| − | |-
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| − | |valign="top"|{{Ref|D}}|| Richard M. Dudley, "Real analysis and probability", Wadsworth&Brooks/Cole (1989). {{MR|0982264}} {{ZBL|0686.60001}}
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| − | |-
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| − | |valign="top"|{{Ref|P}}|| David Pollard, "A user's guide to measure theoretic probability", Cambridge (2002). {{MR|1873379}} {{ZBL|0992.60001}}
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| − | |-
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| − | |valign="top"|{{Ref|I}}|| Kiyosi Itô, "Introduction to probability theory", Cambridge (1984). {{MR|0777504}} {{ZBL|0545.60001}}
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| − | |-
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| − | |valign="top"|{{Ref|M}}||Paul Malliavin, "Integration and probability", Springer-Verlag (1995). {{MR|1335234}} {{ZBL|0874.28001}}
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| − | |-
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| − | |valign="top"|{{Ref|B}}|| V.I. Bogachev, "Measure theory", Springer-Verlag (2007). {{MR|2267655}} {{ZBL|1120.28001}}
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| − | |-
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| − | |valign="top"|{{Ref|F}}|| D.H. Fremlin, "Measure theory", Torres Fremlin, Colchester. Vol. 1: 2004 {{MR|2462519}} {{ZBL|1162.28001}}; Vol. 2: 2003 {{MR|2462280}} {{ZBL|1165.28001}}; Vol. 3: 2004 {{MR|2459668}} {{ZBL|1165.28002}}; Vol. 4: 2006 {{MR|2462372}} {{ZBL|1166.28001}}
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| − | |valign="top"|{{Ref|H}}|| Jean Haezendonck, "Abstract Lebesgue-Rohlin spaces", ''Bull. Soc. Math. de Belgique'' '''25''' (1973), 243–258. {{MR|0335733}} {{ZBL|0308.60006}}
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| − | |valign="top"|{{Ref|S}}|| Zbyněk Šidák, "On relations between strict-sense and wide-sense conditional expectations", ''Teor. Veroyatnost. i Primenen.'' '''2''' (1957), 283–288. {{MR|0092249}} {{ZBL|0078.31101}}
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| − | |}
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