# Georgii Evgen'evich Shilov's scientific contributions

## Citations

... This theorem holds for any two measures and measurable function S. The proof is based on expressing E P as a function of E Q and applying Jensen's inequality. Appendix A contains details. dQ * dP is the Radon-Nikodym derivative, and is the relative density of measure Q * with respect to P (Gurevich et al., 1966). ...
... (cf. ,  Section 10.2) Let (X, F n , µ n ) n∈N and (X, F n , ν n ) n∈N be two sequences of measures on the same space X and same σ-algebras (X, F n ). Suppose that both sequences form a projective system and satisfy Kolmogorov's consistency condition, so that by Lemma 2.13, we have induced measures µ, ν on the σ-algebra F := n F n generated by ∪ n F n . ...
... Let m be a Borel probability measure on X B . Since X B is naturally partitioned into a refining sequence of clopen partitions Q n formed by cylinder sets of length n, we can apply de Possel's theorem (see, for instance, [SG77]). We have that for m-a.a x, ...
... We'd discuss the matters in class together. Hallett's account recalls a course taught by H. H. Mitchell using a real variables textbook written by Yale's James Pierpont . The " other course " in the penultimate sentence is most likely Moore's " Foundations of Mathematics " since he offered it when Hallett matriculated in 1916. ...
... In the second approach, a survival cutoff time was applied by forcing the predicted survival curve to 0 at different time points (i.e., 5, 10 or 15 years) to limit the effect of the long tail of the log-logistic or lognormal distribution. The area under the truncated curve was then calculated using the Riemann sum approximation technique (i.e., dividing the area under a curve into small rectangles, calculating the area of each rectangle, and then summing up the areas to approximate the area under the curve (Shilov et al, 1977)). This second approach is commonly used in economic evaluations when information on maximum survival is lacking and the extrapolated survival curve has a long tail. ...
... The infinite sum ∑ ∞ k=−∞ c k ϕ k is called the Fourier series expansion of f , where {c k } are the Fourier coefficients. Using Equation (29) and taking the inner product of both sides of Equation (30) by ϕ n , for any n ∈ Z, yields c n ...
... The derivatives f (i) , i ∈ N, are defined like the numerical ones, see , p. 83. The integral x y in (46) is of Bochner type, see . ...