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Jun
2
revised Monotone Convergence Theorem for non-negative decreasing sequence of measurable functions
another tag.
Jun
2
revised Monotone Convergence Theorem for non-negative decreasing sequence of measurable functions
Hi. I fixed one point at your answer, namely the claim about the limit (crucial typo there: you wrote f_n instead of f). And also, +1 !
Jun
2
reviewed Approve Limit of a function involving trigonometric exponents.
Jun
2
comment What is the topology here with three elements in sets?
@Topology: The complement of $\{b\}$ is $\{a,c\}$, which is neither an element of $\tau$, nor $\tau$ without $\{b\}$ in it.
Jun
2
answered What is the topology here with three elements in sets?
Jun
2
revised How to show that $\lambda(E)=1$??
edited tags
Jun
2
answered How to show that $\lambda(E)=1$??
Jun
2
comment Compactness of a group with a bounded left-invariant metric
Why is this left invariant? Maybe it's obvious but I'm not seeing it.
Jun
1
reviewed Approve continuity of product path--pasting lemma
Jun
1
reviewed Approve Does the square of uniform distribution have density function?
Jun
1
comment Prove the dominated convergence for $f_n(x)=\frac{x^n}{x^2+3x+2}$
@Lucas: The conclusion is that $\lim_{n\to\infty}\int_{0}^{1}f_{n}(x)=\int_{0}^{1}f(x)$. And $f(x)=0$ if $x\in [0,1)$ and $f(1)=\frac{1}{6}$. This limit function is defined on the whole interval $[0,1]$ and is discontinuous. The reason why $\int_{0}^{1}f(x)=0$ requires measure theory: the set $\{1\}$ has measure zero and thus its contribution to the integral $\int_{0}^{1}f(x)$ is negligible. To fully understand this, a good place to start is to google "Lebesgue measure" and "Lebesgue integral".
Jun
1
comment Prove the dominated convergence for $f_n(x)=\frac{x^n}{x^2+3x+2}$
@lucas: Yes that's true. You can ignore the comment about measure zero: I thought you we're dealing with integration through measure theory, but you can also view the integral as a Riemann integral. That's fine too.
Jun
1
comment Prove the dominated convergence for $f_n(x)=\frac{x^n}{x^2+3x+2}$
@lucas: The point wise convergence to zero happens for all $x\in [0,1)$. At $x=1$ the limit is not zero, but the single point $x=1$ has measure zero and thus does not contribute to the value of the integral.
Jun
1
comment Prove the dominated convergence for $f_n(x)=\frac{x^n}{x^2+3x+2}$
@Lucas: I edited the answer and included that part too. For further reference for dominated convergence theorem, the wikipedia article has a lot of information: en.wikipedia.org/wiki/Dominated_convergence_theorem
Jun
1
revised Prove the dominated convergence for $f_n(x)=\frac{x^n}{x^2+3x+2}$
added 144 characters in body
Jun
1
answered Prove the dominated convergence for $f_n(x)=\frac{x^n}{x^2+3x+2}$
Jun
1
answered If $\left| \frac{z-i}{z-1}\right| = \sqrt2$ and $|z| = \sqrt 5$ and $Im(z)<0$, find $Im(z)$, $Re(z)$
Jun
1
revised Cardinality of the collection of all compact metric spaces
added tags
Jun
1
revised Vitali set of outer-measure exactly $1$.
added tags
Jun
1
revised A problem on properties of Hausdorff space
added 35 characters in body