# Sign up for PolicyPal with referral code MATH88 to get PolicyPal credits

Don’t have an account with PolicyPal yet? You’ll need one to start earning credits! Register with referral code MATH88 to receive $10 credits. PolicyPal is a new insurance portal that has various insurance policies. One of the famous ones is “Gigantiq”, which gives an impressive 2% interest per annum. Once you sign up with PolicyPal, using the referral code MATH88, you will receive P$10 PolicyPal credits. Then, just take part in a few simple activities (such as doing really simple quizzes, and do some of their “challenges” which is just clicking a single button). It took me just 5 minutes to earn another P$5 PolicyPal credits. With P$15 PolicyPal credits, you can then redeem your free $10 FairPrice E-Voucher, as shown below. Throughout the entire process, there is no need to fork out a single cent of your own. Do grab your free$10 FairPrice E-Voucher before the offer ends! Once again, the link to sign up for PolicyPal is https://www.policypal.com/referral/r?c=MATH88.

## Free Entry into Science Centre (In Conversation with … John Edmark)

Students interested in the fusion of math, science and art may be interested to attend this wonderful event. Note: Pre-registration is needed for your free entry into the Science centre.

24 March 2018   10:30 – 12:00

Location: Maxwell Auditorium

The Tinkering Studio @ Science Centre Singapore cordially invites you to interact with John Edmark, inventor, designer and artist who teaches design at Stanford University in Palo Alto, California. His most recent work is a series of animated sculptures called Blooms which endlessly unfolds and animates as it spins beneath a strobe light.

Come join us for this talk where John will share his work with logarithmic spiral structures, Fibonacci numbers, and the Golden Ratio (Ф), and how it all led to his inventing Blooms. His masterful illusions are the results of a marriage between art and mathematics.

RSVP: to Ms Jenny Leong by 18 March 2018.

Pre-registration is required for your complimentary entry into the Science Centre.

## Singapore Math Free Resources for Homeschool

Singapore Math is a popular resource for homeschooling children in the United States and many countries. Due to its challenging questions and unique methodology, Singapore Math prepares children for real math at higher levels.

We have curated some of the best Singapore Math Free Resources on the net:

1. https://thegoodgoatmomma.com/2014/10/17/free-singapore-math-curriculum-resources/
Has a good list of free resources including Curriculum and Tutorials.
2. https://singaporemathsource.com/resources/singapore-math-web-sites/
Some free online content to supplement Singapore Math.
3. https://www.freeeducationalresources.com/singaporemath/Singapore_Math.htm
This is an excellent treasure trove of free Singapore Math Worksheets (PDF).
4. https://www.onlinemathlearning.com/singapore-math.html
Many questions with worked solutions with Singapore Math Model method.
5. http://www.sgtestpaper.com/sgmath/
Includes questions modeled from real Singapore schools’ questions.

Do also read our most popular Singapore Math page to find out what is Singapore Math and what are its benefits.

## f integrable implies set where f is infinite is measure zero

Let $(X,\mathcal{M},\mu)$ be a measure space. Let $f:X\to [0,\infty]$ be a measurable function. Suppose that $\int_X f d\mu<\infty$.

(a) Show that the set $\{x\in X:f(x)=\infty\}\subseteq X$ is of $\mu$-measure 0. (Intuitively, this is quite obvious, but we need to prove it rigorously.)

(b) Show that the set $\{x\in X:f(x)\neq 0\}\subseteq X$ is $\sigma$-finite with respect to $\mu$. i.e. it is a countable union of measurable sets of finite $\mu$-measure.

We may use Markov’s inequality, which turns out to be very useful in this question.

Proof: (a) Let $E_k=\{x\in X:f(x)\geq k\}$, where $k\in\mathbb{N}$. Denote $E_\infty=\{x\in X:f(x)=\infty\}$.

$E_K \downarrow E_\infty$, and $\mu (E_1)\leq\frac{1}{1}\int_X f d\mu<\infty$. (Markov Inequality!)

Then

\begin{aligned}\mu(E_\infty)&=\lim_{k\to\infty}\mu (E_k)\\ &\leq\lim_{k\to\infty}\frac{1}{k}\int f d\mu\ \ \ \text{(Markov Inequality)}\\ &=0 \end{aligned}

Therefore, $\mu(E_\infty)=0$.

(b) Let $S_k=\{x\in X:f(x)\geq\frac{1}{k}\}$, $k\in\mathbb{N}$.

$\{x\in X:f(x)\neq 0\}=\cup_{k=1}^\infty S_k$

Therefore, $\mu(S_k)\leq k\int f d\mu<\infty$, and we have expressed the set as a countable union of measurable sets of finite measure.

Once again, do check out the Free Career Quiz!