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Let \(A_1,A_2,…⊂Ω\) be measurable and \(a_1,a_2,…\geq 0\). Is the function \(f=\sum _{n=1}^∞a_n\ind {A_n}\) measurable?
Let \(a_1,…,a_n,b_1,…,b_n∈ℝ\) and \(A_1,…,A_n,B_1,…,B_n⊂Ω\) such that
Let \(f_1,f_2:Ω→[0,∞]\) and \(f=f_1-f_2\). Does that imply \(f_1=f^+\) and \(f_2=f^-\)?
What, if for all \(x∈Ω\) we have \(f_1(x)=0\) or \(f_2(x)=0\)?
Let \(μ\) be a measure on \(ℝ^d\). Is \(ℝ^d\) \(σ\)-finite with respect to \(μ\)?
What, if \(μ\) is a Radon measure?
Given \(f,f_1,f_2,…:Ω→ℝ\), what does it mean that \(f_n→f\) uniformly?
Find an example of functions \(f,f_1,f_2,…:(0,1)→ℝ\) such that \(f_n→f\) pointwise, but not uniformly.
Find \(f:ℝ→ℝ\) and a closed set \(C⊂ℝ\) such that \(f:C→ℝ\) is continuous but \(f:ℝ→ℝ\) is not continuous in every \(x∈C\).
Can you also find an open set with that property?
What is the Lebesgue integral \(∫f\intd \lmg \) for