Is Such a Function Measurable in the Product Measure Space?

Is Such a Function Measurable in the Product Measure Space? 1

You are missing some hypothesis on $x(t,omega)$. Assuming that $(t,omega) to x(t,omega)$ is measurable you can prove this as follows: let $g_n(t,omega)=g(t,frac i 2^n)$ if $x(t,omega) in [frac i-1 2^n,frac i 2^n)$. Then each $g_n$ is measurable and $g_nto g$ pointwise.

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The semifinite portion of a measure $mu$

I agree with you that $mu_1$ as defined in Royden-Fitzpatrick does not yield a measure. An example that shows this is obtained by considering the measure on $mathbbN$ defined on all subsets $A subseteq mathbb N$ by $$ mu(A) = begincases 0, & textif A = emptyset, 1, & textif A = 0 infty, & textotherwise. endcases $$ For this measure $mu$ we have $mu_1(emptyset) = 0$, $mu_1(0) = 1$ and $mu_1(A) = 0$ for all other $A subseteq mathbbN$, so $mu_1$ is obviously not a measure.The usual definition of the semi-finite version $mu_

m sf$ of a measure $mu$ on $(X,Sigma)$ is given by $$ mu_

m sf(E) = supmu(A) mid A subseteq E text measurable, mu(A) lt infty quad textfor E in Sigma $$ In this definition it is clear that $mu_

m sf$ is non-negative, that $mu_

m sf(emptyset) = 0$ and that $mu(E) =mu_

m sf(E)$ whenever $E in Sigma$ has finite $mu$-measure. Using this, it is not hard to check that $mu_

m sf$ is $sigma$-additive on $Sigma$, hence it is a measure on $(X,Sigma)$. Moreover, $mu_

m sf$ is semi-finite and we have $mu_

m sf = mu$ if and only if $mu$ is semi-finite.One can also show that $mu_

m sf$ and $mu$ have the same integrable functions (up to $mu_

m sf$-null sets) and that $mu_

m sf$ is complete only if $mu$ is complete.

Is Such a Function Measurable in the Product Measure Space? 2

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bounding the measure of a disjoint union under a product measure

What your assumption is saying is that the section$$ E^(x)=y mid (x,y)in E $$ is given by $E^(x)=E_x$. Now, your prblem states that $P=P_1times P_2$ is some product measure, so that we can use the Fubini-Tonelli theorem on the indicator function $1_E$ to conclude$$ P(E)=int 1_E (x,y) dP(x,y) =int int 1_E(x,y)dP_2(y) dP_1(x)=int P_2 (E_x) dP_1(x) geq int 1/2 dP_1(x)=1/2. $$

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Integral of $1$ minus Dirichlet's function

The rationals are a set of Lebesgue measure zero, so they do not impact the value of integrals.Here's how you might see that. Start with an epsilon > 0. You can count the rationals, so put an interval of length epsilon/2 over the first rational. Put an interval of length epsilon/4 over the second rational, etc. You've created a sequence of intervals of total length epsilon that cover the rationals, so they must have measure less than epsilon. But epsilon was arbitrary, so they must have measure zero

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Measure theoretic formulation

A probability mass function of a discrete random variable X displaystyle X can be seen as a special case of two more general measure theoretic constructions: the distribution of X displaystyle X and the probability density function of X displaystyle X with respect to the counting measure. We make this more precise below. Suppose that ( A , A , P ) displaystyle (A,mathcal A,P) is a probability space and that ( B , B ) displaystyle (B,mathcal B) is a measurable space whose underlying -algebra is discrete, so in particular contains singleton sets of B displaystyle B . In this setting, a random variable X : A B displaystyle Xcolon Ato B is discrete, provided that its image is countable. The pushforward measure X ( P ) displaystyle X_*(P) -called a distribution of X displaystyle X in this context-is a probability measure on B displaystyle B whose restriction to singleton sets induces a probability mass function f X : B R displaystyle f_Xcolon Bto mathbb R , since f X ( b ) = P ( X 1 ( b ) ) = [ X ( P ) ] ( b ) displaystyle f_X(b)=P(X^-1(b))=[X_*(P)](b) for each b B displaystyle bin B . Now, suppose that ( B , B , ) displaystyle (B,mathcal B,mu ) is a measure space equipped with the counting measure . The probability density function f displaystyle f of X displaystyle X with respect to the counting measure, if it exists, is the Radon-Nikodym derivative of the pushforward measure of X displaystyle X (with respect to the counting measure), so f = d X P / d displaystyle f=dX_*P/dmu and f displaystyle f is a function from B displaystyle B to the non-negative reals. Hence for any b B displaystyle bin B , we have that P ( X = b ) = P ( X 1 ( b ) ) := X 1 ( b ) d P = displaystyle P(X=b)=P(X^-1(b)):=int _X^-1(b)dP= b f d = f ( b ) , displaystyle int _bfdmu =f(b), demonstrating that f displaystyle f is in fact a probability mass function. When there is a natural order among the potential outcomes x displaystyle x , it may be convenient to assign numerical values to them (or n-tuples in case of a discrete multivariate random variable), and to consider also values not in the image of X displaystyle X . That is, f X displaystyle f_X may be defined for all real numbers and f X ( x ) = 0 displaystyle f_X(x)=0 for all x X ( S ) displaystyle x

otin X(S) as shown in the figure. The image of X displaystyle X has a countable subset on which the probability mass function f X ( x ) displaystyle f_X(x) is one. Consequently, the probability mass function is zero for all but a countable number of values of x displaystyle x . The discontinuity of probability mass functions is related to the fact that the cumulative distribution function of a discrete random variable is also discontinuous. If X displaystyle X is a discrete random variable, then P ( X = x ) = 1 displaystyle P(X=x)=1 means that the casual event ( X = x ) displaystyle (X=x) is certain (it is true in the 100% of the occurrences); on the contrary, P ( X = x ) = 0 displaystyle P(X=x)=0 means that the casual event ( X = x ) displaystyle (X=x) is always impossible. This statement is not true for a continuous random variable X displaystyle X , for which P ( X = x ) = 0 displaystyle P(X=x)=0 for any possible x displaystyle x : in fact, by definition, a continuous random variable can have an infinite set of possible values, and thus the probability that it has a single particular value x is equal to 1 = 0 displaystyle tfrac 1infty =0 . Discretization is the process of converting a continuous random variable into a discrete one.

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Where Can I Find Plastic Injection Molding Manufacturers in China?
Where Can I Find Plastic Injection Molding Manufacturers in China?
This article was originally published on the Sourcing Allies blog“What kind of materials can be used in plastic injection molding?”‍PLASTIC INJECTION MOLDINGWhat is the plastic injection molding process?This process is similar to that of metal die casting. In this process, molten plastic is injected into a die under pressure and held until it cools. Once it becomes solid, the die is opened and the plastic casting is taken out. Plastic injection molding is an extremely versatile, precise, fast and inexpensive method of mass producing plastic products.What materials can be used?This process mainly uses both thermoplastics and thermoset plastics, each of which have their own advantages and limitations. The primary difference between the two is that thermoplastics can be melted back into liquid form or recycled while thermoset plastics cannot.ThermoplasticsPros: Recyclable, cost efficient (because waste during manufacturing process can be reused), increased impact resistance because of flexibility.Cons: Lower heat and chemical resistance (they degrade faster.)ThermosetPros: Higher chemical and thermal resistance and higher mechanical strength and dimensional stability.Cons: Cannot be recycled, lower impact resistance. Also, thermoset plastics are more difficult to work with so you might find it difficult to find suppliers for them.What will I need in terms of design input?We must have 3D files in STEP or IGES formats.Where in China can you find plastic injection molding factories?Depending on what kind of plastic you want to use for your product, these suppliers can be found all around China. Different factories have different quality levels and expertise.How much does tooling cost?It can cost from a few thousand dollars for simple tooling to hundreds of thousands for more complex multi-cavity tools.How long does tooling last?It depends on the material, but most likely 200,000 pieces is not a problem.What is the average lead time for tooling?15 days to three months depending on complexity.Do plastic injection molding products or their tooling need any final finishes?It is not unusual for injected parts to need final machining. This may also include addition of inserts (that can also be inserted during casting) or threads cutting. Final surface finishes may also be needed such as powder coating or paint application. Sometimes, under some circumstances the tool itself may need a special surface texture added, which will imprint on the final injected parts.What are the pros and cons of plastic injection molding?Pros:Consistent measurements.Quick production rate.Low cost per unit.Good surface finish.Products can often be recyclable.Manufacturing process can be quite cheap because scrap can be recycled/reused.ConsTooling cost.Warpage and shrinkage.Some thermosetting plastics or rubbers cannot be recycled.Recycled material will not give as good performance as new material.What are the known challenges of the plastic injection molding process?Some suppliers will use recycled material without telling you. Material type used is critical if you are looking for specific physical properties or performance. Additionally, shrinkage can be problematic across larger surface areas.Sourcing Allies is a team of expert China sourcing agents that has helped western customers manufacture and source products from low-cost regions since 2006.For more on plastic injection molding visit our website or write to us at info@sourcingallies.com.·RELATED QUESTIONWhat are some good skin whitening masks?Skin color is not a determiner of someone’s beauty. Having white skin does not mean you are beautiful. On the other hand, you can be an attractive guy with brown or black skin.Let's come to your question. We can't change our skin color through whitening. But we can lighten our skin to increase the glow. There are so many reasons by which we lose our actual skin tone like sunburn, pollution, and chemicals used in cosmetics. Here are some of the Natural face masks that will help you return your actual complexion.1. Turmeric Mask2. Tomato Mask3. Lentil Scrub4. Kiwi Mask5. Potato Mask6. Changing Eating habits.These are the most effective lightening masks for the face. To know more detail about these face masks, I would suggest checking out this resource:Top 10 Face Masks for Skin Lightening.Thanks!.
How to Curl Your Hair Like This?
How to Curl Your Hair Like This?
how to curl your hair like this?To get curls like that you use a curling wand. Almost like an iron but without a clamp and it gets smaller at the bottom. To get bigger curls, you use a thicker wand. To get them not so tight and loose like the picture, you only hold your hair on the wand for 3-4 seconds (different depending on if your hair is fine, curly etc). But you can mess around and look up a tutorial on youtube (there's tons). Good luck!— — — — — —shouldnt blowdrying your hair make it dry?Yes, but it depends on what type of hair you have some people have oily hair but you can not really see the oil unless they blow dry or straighten it, and some people put hair lotions in there hair before the videos or its just left over hair lotions from past days. Other then that yes it does dry your hair blow drying it is not as worse as straightening your hair straightening your hair is cause lots of DAMAGE especially if you do not use heat protection hair sprays/lotions— — — — — —Is your hair long or short???my hair is:: long curly/wavy thick (too thick) natural blond, but i added a little shade hair down with a headband— — — — — —What is better for your hair?I would trim it first (your ends) and then use a flat iron because the blow drier is actaully more damaging because you go over the same spot a lot which is more damaging then one run with a flat iron— — — — — —is TRESemme bad for your hair?Umm, all shampoos are somewhat like a detergant. The trick is you need to put conditioner in your hair after yiu shampoo so you do not look like an afro puff coming out the shower. That's your problem. When you are rinsing out the conditioner, you need to use a brush and brush your hair at the same time while your rinsing the conditioner out. It also depends on your hair texture too and whether or not your hair can handle certain shampoos and conditioners— — — — — —Is dying your hair twice bad?death it to the black is bigger than going decrease back over it with blonde- so jointly because it could make your hair particularly comfortable, it wo not completely fry it or something. omit it after with some deep conditioning remedies— — — — — —How to get your hair grow?Nothing really makes your hair grow faster. On average, hair grows 1/2 per month, which is 6 inches per year— — — — — —what do you die your hair with?Great revelation huh?— — — — — —What REALLY helps your hair grow ?????I understand cutting your hair in anger, i have done it a number of times, and then cried about it for weeks. I take a vitamin that is called Biotin. It is s high-potency B-Vitamin. I had one of my hair stylists tell me about them, and they really do help. You probably wo not notice it much the first month seeing as how it must get into your system but back in the middle of January i had highlights put in my hair and by the end of feburary the roots had grown out an 1 1/2. No joke. So try them and see if they work for you. You can get them at any place that sell vitamins but i usually get mine from GNC, they might cost a little more but they are better for the buy. I would suggest at the least taking 2500 mcg a day. Good luck!.— — — — — —If you put gel in your hair everyday does it ruin your hair?Eventually it will, it is a chemical. I used to use gel and hairspray all the time and my hair was rough and nasty. I have not used any of that in a couple years and my hair is beautiful now— — — — — —How to deep condition your hair?Here is a simple organic inexpensive recipe I use: 1. Crack a raw egg into a small bowl. 2. Add 2 tbsp, of olive oil as well 3. Blend the two together with a fork for about 1 minute. 4. Massage treatment into your scalp and saturate your hair with the mixture. 5.Place a plastic shower cap on your hair for 15 minutes 6. Rinse in cool water thoroughly (Do not use Hot water) Deep conditioners add protein to repair your hair, Eggs are a rich source of protein that works well.
What Is Preventing the Solar Energy and Other Renewables to Replace the Oil?
What Is Preventing the Solar Energy and Other Renewables to Replace the Oil?
Solar energy and other renewables do not return the energy invested compared with fossil fuels. Fossil fuels are also portable-they have lots of energy and they can be transported. Not true of solar or wind, which are site specific and have to be part of an electrical grid. There's also the fact that the oil and coal still sitting underground is worth trillions to the oil companies, and the people who run those companies get rich from the status quo1. what is the most common and renewable energy sources?solar energy or w can use water as hydropowr2. Do solar energy panels for your home make sense?I looked into it once, just for the hell of it. I assumed the sun would shine 12 hours every day, which it does not in this part of the world.I assumed no maintenance cost. The pay-back was about 20 years.In all the companies I have worked for, if the pay-back on a project is over 4 years, it has no chance. The payback depends on the cost of electricity from the local supplier, the maintenance cost, the amount and intensity of sunlight and the installed cost of panels. This is all constantly changing. As the situation stands now, I would rather invest my money in lottery tickets or better still buy good stocks.3. What kind of fuel is the most used one for generating energy?We rely mainly on coal, nuclear, natural gas, hydroelectric, and petroleum power and a small amount from solar energy, tidal harnesses, wind generators, and geothermal sources. 80% of the world's energy consumption is in fossil fuels -- coal, oil and natural gas. Coal is still "king" right now at almost 50% of the US energy needs. Oil is a modest second. .4. How much do you know about solar energy?enough to know it's not the way to go for the future. I also know enough not to count wikipedia as a reliable resource. I firmly believe that nuclear is going to be the common source of power in the future.5. How is solar energy stored for later use?I have a solar panel system installed by a team called cotterill civils from birmingham, the midlands. If you need understand more about it, i would see there site as it has an informational6. Solar Energy | Save & Learn More with ChooseEnergy®The United States is on track to top 3 million solar PV installations this year, adding 1 million photovoltaic panels in just two years, according to a study by Wood Mackenzie Power & Renewables and the Solar Energy Industries Association. The number will exceed 4 million in 2023, the study says. It took more than 40 years for the first million installations. "The rapid growth in the solar industry has completely reshaped the energy conversation in this country," said Abigail Ross Hopper, SEIA president and CEO, in a press release announcing the study. The 2.7 million solar systems installed through September 2020 have the capacity to generate nearly 89 gigawatts of electricity. The solar industry employs about 250,000 Americans, the SEIA says. The expansion of solar is part of a growing movement toward renewable energy. Green energy made up 22.5 percent of all electricity generated in the country in 2020, according to the U.S. Energy Information Administration. Solar energy was responsible for about 14.7 percent of the green energy generated.7. can solar energy generate electricity in a home alone? what is needed?.?Adequate #of Solar Panels Adequate #of Storage Batteries Battery Charge Controller System load management Controller Mounting and wiring equipment Gas or Diesel Generator (Properly sized) I would figure on spending around $30,000-$40,000 for a decent system. This system will not give you air conditioning or heat, but will probably be adequate for a small homes other electrical, with some inconveniences at times. There are many companies offering these on the Internet. Some of them have forms and calculations for sizing all the equipment based on your location. You are not in a very good area because of a lot of rain, so diesel will need to run a lot. The most important thing in sizing system is to be accurate in listing ALL appliances electrical loads you have and how many hours/day they operate.
What Matches This Formal Dress?
What Matches This Formal Dress?
straight hair silver jewerly..and really try to find silver shoes because they would definatly look best..have fun!1. Great hairstyle for a teen girl with long straight hair?cornrows,twist,braids in the front up in ponytail,french braids,you can even cover the eye when it is out Im an hairstylist2. My hair is outrageously curly and frizzy, help!?Your hair texture changes as you age due to hormones. The stylists will recommend a keratin treatment - this contains formaldehyde, which is carcinogenic. Do not believe them if they say it is formaldehyde-free or chemical-free, or that XYZ brand is 'safe' - the process requires formaldehyde to set the keratin, and the heat changes some ingredients to formaldehyde gas. If the stylist needs a mask in order to put it on you, it's toxic. Some women have also had all their hair fall out from keratin treatments. Curly hair needs different techniques to straight hair to maintain - you are more than your hair, but working with your hair's natural texture is easier than working against it.3. what is better [natural] curly or straight hair?Curly. And I dispute most of the thing you said that you can not do with it. Sure, it's not the same as with the straight, but it's not impossible. As for myself, I have had bangs, and well, duh you can not brush it without frizzing, you are separating the individual strands. And I find it entertaining to try new hair styles and ways of pinning hair up, far more than just putting gel in. Sounds to me like learning more about curly hair would solve most of your problems with4. What's IN and what's OUT?Skinny headbands-IN :) Mascarra- in Lip Gloss- neither Black nailpolish- in Red nailpolish- out Messy bun- in High ponytail- in Juicy Couture sweatshirts- out Juicy Couture sweatpants- in UGG Australia boots- in but working its way out Coach sneakers- out The North Face sweatshirts- in True Religion Jeans- neither Abercrombie & Fitch- out Hollister Co.- in Aeropostale- in American Eagle- in Forever 21- in Lacoste- out Ralph Lauren- ehh kind of both Old Navy- in The Gap- out iPhone- out Skinny jeans- in Bootcut jeans- out Ripped jeans- in Open toed heals- in Pumps- in Betsy Johnson-out Lily Pulitzer-in Tommy Hilfigure-out Gossip Girl-in The Hills-out The City-out Britney Spears-in Lady Gaga-in Rihanna-in Chris Brown-OUTOUTOUT Jonas Brothers-in Miley Cyrus-in Demi Lovato-in Selena Gomez-in Disney Chanel-in Pearl earings-in Diamond earings-in Curly hair-in Straight hair-in Short hair-in Shoulder length hair-out Front bangs-in Side bangs-in Red lipstick-out Mini skirts-in Aviators-in Raybans-out Oversized sunglasses-neither Facebook-in Myspace-in5. What can i do with my straight hair?i have long straight dark brown hair i usually tease my hair to give it some volume to be cute and straiten it to make it extra straight. I also have clip in extensions that i use everyday just to make my hair less boring there purple and i put them on my bottom layer of hair, with my black under color. i hope i helped6. Is there any way to get straight hair permanently?No, there's no way whatsoever to permanently change your hair texture. All you can do it get long term treatments like chemical straightening. But even that grows out and your roots will be your natural texture.7. how do u curl shoulder length asian hair with a straightener?sorry ms kimchi how can you curl it with a straightener when you already have stick straight hair. Go buy a curling iron it's only $208. How to curl wiry thick and pin straight hair?Its the gene code. The dna strands tell your hair whether to be made curly or straight. It is genetics. Not "besides genetics", thats all there is9. What are cute hairstyles with NO HEAT? For straight hair?You can half of your hair up and half down. That's always cute10. my hair for the spring formal?Straight hair would look good with your dress just have some body in it do not wear it flat to your head and do not get zebra tips on your nails that does not go with that dress. TRUST ME..Get a french manicure or get your nails polished the same color as the dress
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