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This thesis attempts to prove the Lebesgue measure is a concrete realization of measure.

The measure-multiplicity-invariant for masas in II_1 factors was introduced by Dykema, Smith and Sinclair to distinguish masas that have the same Pukanszky invariant. In this dissertation, the measure class (left-right-measure) in the measuremultiplicity- invariant is studied, which equivalent to studying the structure of the standard Hilbert space as an associated bimodule. The focal point of this analysis is: To what extent the associated bimodule remembers properties of the masa. The structure of normaliser of any masa is characterized depending on this measure class, by using Baire category methods (Selection principle of Jankov and von Neumann). Measure theoretic proofs of Chifan's normaliser formula and the equivalence of weak asymptotic homomorphism property (WAHP) and singularity is presented. Stronger notions of singularity is also investigated. Analytical conditions based on Fourier coefficients of certain measures are discussed, that partially characterize strongly mixing masas and masas with nontrivial centralizing sequences. The analysis also provide conditions in terms of operators and L2 vectors that characterize masas whose left-right-measure belongs to the class of product measure. An example of a simple masa in the hyperfinite II1 factor whose left-right-measure is the class of product measure is exhibited. An example of a masa in the hyperfinite II1 factor whose leftright- measure is singular to the product measure is also presented. Unitary conjugacy of masas is studied by providing examples of non unitary conjugate masas. Finally, it is shown that for k greater than/equal to 2 and for each subset S \subseteq N, there exist uncountably many non conjugate singular masas in L(Fk) whose Pukanszky invariant is S u {1}.

In 2 ¡Ñ2 case,we discuss the uniqueness of the u-invariant measure on projective space.Under the condition that |detM|=1 for any M in Gu and Gu is not compact,we have the followings: (1) For any x in P(R^2),if #{M¡Dx|M belongs Gu}>2, then the u-invariant measure is unique. (2) For some x in P(R^2),there exists x1,x2 such that {M¡Dx|M belongs Gu} is contained in {x1,x2},if x1 and x2 are both fixed,then the u-invariant measure v is not unique;otherwise,if u has mass only on x1 and x2,then the u-invariant measure is unique.

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This dissertation is divided into two parts. The first part addresses the following problem: Suppose 𝑣 is a finitely additive probability measure defined on the power set 𝒜 of the integer Z so that each singleton set gets measure zero. Let X be a product space Π/β∈B * Zᵦ where each Zₐ is a copy of the integers. Let 𝒜ᴮ be the algebra of subsets of X generated by the subproducts Π/β∈B * Cᵦ where for all but finitely many β, Cᵦ = Zᵦ. Let 𝑣_B denote the product measure on 𝒜ᴮ which has each factor measure a copy of 𝑣. A subset E of X is said to be 𝑣_B -measurable iff [sic] there is only one finitely additive probability on the algebra generated by 𝒜ᴮ ∪ [E] which extends 𝑣_B. The set E ⊆ X is said to be universally product measurable (u.p.m.) iff [sic] for each finitely additive probability measure μ on 𝒜 which gives each singleton measure zero,E is μ_B -measurable. Two theorems are proved along with generalizations. The second part of this dissertation gives a proof of the following theorem and some generalizations: There are 2ᶜ nonisomorphic subalgebras of the power set algebra of the integers (where c = power of the continuum).

The current dissertation is a replication of a pilot study and aims to define what it means for group leaders to "act on" feedback from a group therapy process measure called the Group Questionnaire (GQ). Twelve leaders received feedback reports based on group member responses to the GQ after each session. Leaders submitted two sources of qualitative data: brief written session-by-session explanations of feedback use and end-of-semester debrief interviews to describe their experience with the measure. Researchers conducted a qualitative content analysis that yielded 15 categories of leader GQ use summarized by three temporal dimensions. Quantitative analyses were performed to test for variability in leader use. Categories common to both the pilot study and the current study were established, and the two data sets were merged to create one complete data set. A brief questionnaire designed to summarize leader use is introduced, and quantitative analyses were performed to test the relationship between this measure and qualitative findings. Finally, implications of these findings are discussed.

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