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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

2001 • Question 5

Question

Math 6210 • Real Analysis 2001 • Question 5

Prove that

(a) \(\log (1+e^t)\le t+\log (2)\) for all \(t\ge 0\);

(b) if \(f\) is a real valued function in \(L^1([0,1])\) then \[ \lim _{n\to \infty }\frac {1}{n}\int _0^1 \log (1+e^{n f(x)})\,dx =\int _0^1 f^+(x)\,dx, \] where \(f^+(x)=f(x)\) if \(f(x)\ge 0\) and \(f^+(x)=0\) if \(f(x)<0\).

Raw LaTeX

Prove that

(a) \(\log(1+e^t)\le t+\log(2)\) for all \(t\ge 0\);

(b) if \(f\) is a real valued function in \(L^1([0,1])\) then
\[
\lim_{n\to\infty}\frac{1}{n}\int_0^1 \log(1+e^{n f(x)})\,dx
=\int_0^1 f^+(x)\,dx,
\]
where \(f^+(x)=f(x)\) if \(f(x)\ge 0\) and \(f^+(x)=0\) if \(f(x)<0\).

Topic Tags

Lebesgue Integration Lp Spaces and Sequence Spaces