Taylor Martin
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 Dec3 comment Proving uniform convergence of an integral-defined function on compact sets Ah, then you will have to adjust the limits in the integral accordingly to take into account the domain. But note for instance that if $f$ is supported on $K\subset\mathbb{R}$, then $f_{\epsilon}$ is supported on a subset $K_{\epsilon}\subset K$. The specific range of $f$ is irrelevant, only that it allows for $\int_{\mathbb{R}} f=1$ and that $f\geq0$. Dec2 revised Proving uniform convergence of an integral-defined function on compact sets added 939 characters in body Dec2 comment Proving uniform convergence of an integral-defined function on compact sets Actually, since uniform convergence is usually harder to demonstrate than pointwise convergence, a good starting point would be to prove pointwise convergence first, even if restricted to compact sets. I elaborated on my answer a bit; hopefully it will get you started. Dec2 revised If $\lim_{n \to \infty} f_n=f$ (Almost everywhere) then $\lim_{n \to \infty} f_n=f$ ( in measure on$E$) added 275 characters in body Dec1 revised $y=ce^{y/x};\quad y'=y^2/(xy-x^2)$ added 250 characters in body Dec1 answered $y=ce^{y/x};\quad y'=y^2/(xy-x^2)$ Dec1 revised If $\lim_{n \to \infty} f_n=f$ (Almost everywhere) then $\lim_{n \to \infty} f_n=f$ ( in measure on$E$) added 141 characters in body Dec1 answered Proving uniform convergence of an integral-defined function on compact sets Dec1 answered If $\lim_{n \to \infty} f_n=f$ (Almost everywhere) then $\lim_{n \to \infty} f_n=f$ ( in measure on$E$) Dec1 comment Proving uniform convergence of an integral-defined function on compact sets In the statement of your problem, does $f$ have compact support? Dec1 comment Orders of growth of typical sequences Are you using the $\ll$ symbol to mean $P(n)\ll Q(n)$ if there is an absolute constant $C$ such that $P(n)\leq CQ(n)$ or that $P(n)/Q(n)\leq C$ for $n\geq N$ for some $N$ sufficiently large; or is it notation for one quantity being "much smaller" than the other? Also, are the numbers $p_{j},q_{j}$ and $a_{j},b_{j}$ arbitrary pairs with no relation between the indices? Nov20 comment Sum to infinity of the sum 1/n^2 Look to the right as you type this question and you'll see a ton of similar questions w/ answers. There's even one with literally dozens of solutions. Nov17 answered Equivalency of Norms and the Open Mapping Theorem Nov4 comment Resource on Pathwise Computations Involving Brownian Motion Thanks - I edited my question to take some of this into account. Nov4 revised Resource on Pathwise Computations Involving Brownian Motion added 1732 characters in body Nov4 accepted Resource on Pathwise Computations Involving Brownian Motion Nov4 revised Resource on Pathwise Computations Involving Brownian Motion added 1732 characters in body Oct30 answered Solution to $\frac{d^2 y}{dt^2}+y=\sec\left(t\right)$ Oct28 asked Resource on Pathwise Computations Involving Brownian Motion Oct26 comment Proving that if $f$ is Riemann integrable and $1/f$ is bounded then $1/f$ is Riemann integrable If you wanted to make it all workout to $\epsilon$ at the end, then just refine $P$ (if necessary) so that $|U(P,f)-L(P,f)|<(m_{f})^{2}\epsilon.$ But understand that this is not necessaty, since in general a quantity bounded by $C\epsilon$ for a constant that does not depend on $\epsilon$ is as good as being bounded by $\epsilon$ since both upper bounds can be made arbitrary small by sending $\epsilon\to0$.