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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2007 • Question 5

Question

Math 6210 • Real Analysis Fall 2007 • Question 5

Let \(I=[0,1]\) be the unit interval with the Lebesgue measure, and let \(I\times I\) be endowed with the product measure. Assume that \(K\in L^2(I\times I)\) and \(f\in L^2(I)\). Define \[ g(x)=\int _0^1 K(x,y)f(y)\,dy \] Show that \(g\in L^2(I)\).

Raw LaTeX

Let \(I=[0,1]\) be the unit interval with the Lebesgue measure, and
let \(I\times I\) be endowed with the product measure. Assume that
\(K\in L^2(I\times I)\) and \(f\in L^2(I)\). Define
\[
g(x)=\int_0^1 K(x,y)f(y)\,dy
\]
Show that \(g\in L^2(I)\).

Topic Tags

Lebesgue Integration Lebesgue Measure and Translation Invariance Lp Spaces and Sequence Spaces Product Measures and Fubini-Tonelli Theorems