छ्येलेमि:Eukesh/Auto/NepalScript/दाये (गणित)

थ्व पौ देवनागरीं बोट छ्यला थःमंतुं नेपाललिपिइ हिलातःगु ख। थ्व पौयात च्वसु पतिइ यंकेन्ह्यः छकः ब्वनाः पाय्छि जू/मजूगू स्वयादिसँ।

Informally, a measure has the property of being monotone

alt=|thumb| 𑐀𑐣𑑁𑐥𑐔𑐵𑐬𑐶𑐎𑐎𑐠𑑄, 𑐡𑐵𑐫𑐾𑐐𑐸𑐫𑐵 𑐩𑑀𑐣𑑀𑐚𑑀𑐣 𑐖𑐸𑐂𑐐𑐸 𑐐𑐸𑐞 𑐡𑐂 𑐠𑐸𑐐𑐸 𑐀𑐬𑑂𑐠𑐫𑑂 𑐎𑐶 𑐫𑐡𑐶 $A$ 𑐠𑐣 $B,$𑐫𑐵 𑐕𑐐𑐹 𑐄𑐥𑐥𑐸𑐔𑑅 𑐏𑑅 𑐢𑐵𑑅𑐳𑐵 𑐡𑐵𑐥𑐹 $A$𑐫𑐵 𑐡𑐵𑐥𑐹 $B.$𑐳𑑂𑐰𑐫𑐵𑑄 𑐩𑑂𑐴𑑂𑐰 𑐧𑐵𑐫𑑂 𑐧𑐬𑐵𑐧𑐬 𑐖𑐸𑐂। 𑐠𑑂𑐰𑐫𑐵𑑄 𑐀𑐥𑑂𑐰𑑅, 𑐏𑐵𑐮𑐶𑐐𑐸 𑐳𑐾𑐚𑐫𑐵 𑐡𑐵𑐥𑐹 𑑐 𑐖𑐸𑐂𑐩𑐵𑑅 । 𑐡𑐵𑐫𑐾𑐐𑐸 𑐖𑑂𑐫𑐵𑐫𑐵 𑐕𑐐𑐹 𑐳𑐵𑐢𑐵𑐬𑐞 𑐡𑐳𑐸 𑐢𑐫𑐵𑐐𑐸 𑐁𑐫𑐟𑐣 (𑐕𑐸𑑄 𑐧𑐳𑑂𑐟𑐸𑑄 𑐐𑐸𑐮𑐶 𑐟𑑅𑐢𑑄𑐐𑐸 𑐠𑐵𑐫𑑂 𑐎𑐵𑐂) 𑐏𑑅 । 𑐐𑐞𑐶𑐟𑐫𑑂 𑐡𑐵𑐫𑐾 𑐧𑐵𑐫𑑂 𑐡𑐵𑐫𑐾𑐐𑐸𑐫𑐵 𑐀𑐰𑐢𑐵𑐬𑐞𑐵 𑐖𑑂𑐫𑐵𑐩𑐶𑐟𑐷𑐫 𑐡𑐵𑐫𑐾 ( 𑐴𑐵𑐎𑑅, 𑐎𑑂𑐲𑐾𑐟𑑂𑐬𑐦𑐮, 𑐁𑐫𑐟𑐣 ) 𑐰 𑐩𑐾𑐩𑐾𑐐𑐸 𑐳𑐵𑐩𑐵𑐣𑑂𑐫 𑐀𑐰𑐳𑑂𑐠𑐵, 𑐡𑐳𑐸 𑐥𑐬𑐶𑐩𑐵𑐞, 𑐡𑑂𑐬𑐰𑑂𑐫𑐩𑐵𑐣, 𑐰 𑐑𑐚𑐣𑐵𑐫𑐵 𑐳𑐩𑑂𑐨𑐵𑐰𑐣𑐵𑐫𑐵 𑐍𑐥𑐔𑐵𑐬𑐶𑐎 𑐩𑑂𑐴𑐳𑐷𑐎𑐵𑐫𑐵 𑐎𑐸𑐟𑑅 𑐏। 𑐠𑑂𑐰 𑐧𑐶𑐳𑑂𑐎𑑄 𑐠𑐾𑑄 𑐏𑐣𑐾𑐡𑐸𑐐𑐸 𑐠𑐷𑐠𑐷 𑐀𑐰𑐢𑐵𑐬𑐞𑐵𑐫𑑂 𑐫𑐎𑑂𑐰 𑐳𑐩𑐵𑐣𑐟𑐵 𑐡𑐸 𑐰 𑐥𑑂𑐬𑐵𑐫𑑅 𑐕𑐐𑐹 𑐴𑐾 𑐐𑐞𑐶𑐟𑐷𑐫 𑐳𑐣𑑂𑐡𑐬𑑂𑐨𑐫𑑂 𑐕𑐠𑐵𑐫𑑂 𑐟𑐫𑐵𑑅 𑐰𑑂𑐫𑐰𑐴𑐵𑐬 𑐫𑐵𑐫𑐾𑐕𑐶𑑄। 𑐡𑐵𑐫𑐾𑐐𑐸 𑐳𑐩𑑂𑐨𑐵𑐰𑑂𑐫𑐟𑐵 𑐳𑐶𑐡𑑂𑐢𑐵𑐣𑑂𑐟, 𑐊𑐎𑐷𑐎𑐬𑐞 𑐳𑐶𑐡𑑂𑐢𑐵𑐣𑑂𑐟𑐫𑑂 𑐁𑐢𑐵𑐬𑐨𑐹𑐟 𑐀𑐰𑐢𑐵𑐬𑐞𑐵 𑐏 𑐰 𑐰𑐶𑐡𑑂𑐫𑐸𑐟 𑐔𑐵𑐬𑑂𑐖 𑐠𑐾𑑄 𑐆𑐞𑐵𑐟𑑂𑐩𑐎 𑐩𑐵𑐟𑑂𑐬𑐵 𑐐𑑂𑐬𑐴𑐞 𑐫𑐵𑐫𑐾𑐟 𑐣𑑄 𑐳𑐵𑐩𑐵𑐣𑑂𑐫𑐷𑐎𑐬𑐞 𑐫𑐵𑐫𑐾𑐕𑐶𑑄। 𑐡𑐵𑐫𑐾𑐐𑐸𑐫𑐵 𑐩𑐾𑐩𑐾𑐐𑐸 𑐟𑐵𑑅𑐥𑐵𑑅𑐐𑐸 𑐳𑐵𑐩𑐵𑐣𑑂𑐫𑐷𑐎𑐬𑐞 (𑐡𑐳𑐸 𑐳𑑂𑐥𑐾𑐎𑑂𑐟𑑂𑐬𑐮 𑐡𑐵𑐫𑐾 𑐰 𑐥𑑂𑐬𑐎𑑂𑐲𑐾𑐥𑐞-𑐩𑐹 𑐡𑐵𑐫𑐾 ) 𑐎𑑂𑐰𑐵𑐣𑑂𑐚𑐩 𑐨𑑁𑐟𑐶𑐎 𑐰𑐶𑐖𑑂𑐘𑐵𑐣 𑐰 𑐳𑐵𑐩𑐵𑐣𑑂𑐫 𑐬𑐸𑐥𑐫𑑂 𑐨𑑁𑐟𑐶𑐎 𑐰𑐶𑐖𑑂𑐘𑐵𑐣𑐫𑑂 𑐰𑑂𑐫𑐵𑐥𑐎 𑐬𑐸𑐥𑑄 𑐕𑑂𑐫𑐾𑐮𑐶𑐐𑐸 𑐫𑐵।

𑐠𑑂𑐰 𑐀𑐰𑐢𑐵𑐬𑐞𑐵𑐫𑐵 𑐮𑑂𑐫𑐹𑐣𑐾𑐫𑐵 𑐀𑐣𑑂𑐟𑐬𑑂𑐖𑑂𑐘𑐵𑐣 𑐥𑑂𑐬𑐵𑐔𑐷𑐣 𑐐𑑂𑐬𑐷𑐳𑐫𑐵 𑐃𑐫𑐵 𑐏𑑅, 𑐐𑐸𑐧𑐮𑐫𑑂 𑐁𑐬𑑂𑐎𑐶𑐩𑐶𑐜𑐷𑐖𑑄 𑐕𑐐𑐹 𑐔𑐵𑐎𑑅𑐫𑐵 𑐎𑑂𑐲𑐾𑐟𑑂𑐬𑐦𑐮𑐫𑐵 𑐮𑑂𑐫𑐵𑐏𑐵𑐫𑐾𑐟 𑐳𑑂𑐰𑑅𑐐𑐸 𑐏𑑅। 𑐟𑐬 𑑑𑑙𑐐𑐸 𑐱𑐟𑐵𑐧𑑂𑐡𑐷𑐫𑐵 𑐀𑐣𑑂𑐟 𑐰 𑑒𑑐𑐐𑐸 𑐱𑐟𑐵𑐧𑑂𑐡𑐷𑐫𑐵 𑐳𑐸𑐬𑐸𑐂 𑐖𑐎 𑐡𑐵𑐫𑐾 𑐳𑐶𑐡𑑂𑐢𑐵𑐣𑑂𑐟 𑐐𑐞𑐶𑐟𑐫𑐵 𑐕𑐐𑐹 𑐎𑐔𑐵 𑐖𑐸𑐮। 𑐁𑐢𑐸𑐣𑐶𑐎 𑐡𑐵𑐫𑐾 𑐳𑐶𑐡𑑂𑐢𑐵𑐣𑑂𑐟𑐫𑐵 𑐖𑐐 𑐊𑐩𑐶𑐮 𑐧𑑀𑐬𑐾𑐮, 𑐴𑐾𑐣𑐬𑐷 𑐮𑐾𑐧𑐾𑐐, 𑐣𑐶𑐎𑑀𑐮𑐵𑐃 𑐮𑐸𑐖𑐶𑐣, 𑐖𑑀𑐴𑐵𑐣 𑐬𑐾𑐜𑐣, 𑐎𑐣𑑂𑐳𑑂𑐚𑑂𑐫𑐵𑐣𑑂𑐚𑐶𑐣 𑐎𑑂𑐫𑐵𑐬𑐵𑐠𑐶𑐫𑑀𑐜𑑀𑐬𑐷, 𑐰 𑐩𑑁𑐬𑐶𑐳 𑐦𑑂𑐬𑐾𑐔𑐾𑐚 𑐁𑐡𑐶𑐫𑐵 𑐖𑑂𑐫𑐵𑑄 𑐡𑐫𑐾𑐎𑐹𑐐𑐸 𑐏।

𑐠𑐸𑐂𑐎𑐾

thumb|300x300px| 𑐡𑐵𑐥𑐹𑐫𑐵 𑐮𑑂𑐫𑐵𑑅 𑐏𑐵𑐫𑐾 𑐖𑑂𑐫𑐹𑐐𑐸 𑐫𑑀𑐐𑐱𑐷𑐮𑐟𑐵 $\mu$ : 𑐡𑐵𑐫𑐾𑐖𑑂𑐫𑐹𑐐𑐸 𑐀𑐳𑐩𑑂𑐧𑐡𑑂𑐢 𑐳𑑄𑐑𑐫𑐵 𑐡𑐵𑐫𑐾 𑐡𑐎𑑂𑐰𑑄 𑐄𑐥𑐥𑐸𑐔𑑅𑐫𑐵 𑐳𑐎𑐮 𑐮𑑂𑐫𑐵𑐫𑐵 𑐫𑑀𑐐 𑐖𑐸𑐂। $X$ 𑐕𑐐𑐹 𑐳𑐾𑐚 𑐖𑐸𑐳𑐵 𑐀𑐮𑐾 $\Sigma$ a 𑐕𑐐𑐹 σ-𑐀𑐮𑑂𑐖𑐾𑐧𑑂𑐬𑐵 𑐌𑐨𑐬 $X.$। $\Sigma$ 𑐣𑑄 𑐣𑑂𑐴𑑂𑐫𑐠𑑄𑐐𑐸 𑐰𑐶𑐳𑑂𑐟𑐵𑐬𑐶𑐟 𑐰𑐵𑐳𑑂𑐟𑐰𑐶𑐎 𑐮𑑂𑐫𑐵𑑅𑐫𑐵 𑐢𑑂𑐰𑑅𑐫𑐵 𑐕𑐐𑐹 𑐳𑐾𑐚 𑐦𑑄𑐎𑑂𑐳𑐣 $\mu$ 𑐫𑐵𑐟 𑐡𑐵𑐫𑐾 𑐢𑐵𑐂 𑐫𑐡𑐶 𑐎𑑂𑐰𑐫𑑂 𑐧𑐶𑐫𑐵𑐟𑑅𑐐𑐸 𑐱𑐬𑑂𑐟𑐟 𑐥𑐵𑐮𑐣𑐵 𑐖𑐸𑐳𑐵:

• 𑐐𑐿𑐬-𑐣𑐎𑐵𑐬𑐵𑐟𑑂𑐩𑐎𑐟𑐵 : 𑐳𑐎𑐳𑐶𑐟𑑄 $E\in \Sigma ,\ \ \mu (E)\geq 0.$
• $\mu (\varnothing )=0.$
• 𑐮𑑂𑐫𑐵𑑅 𑐏𑐵𑐫𑐾 𑐖𑑂𑐫𑐹𑐐𑐸 𑐫𑑀𑐖𑐎𑐟𑐵 (𑐧𑐵𑐫𑑂 σ-𑐫𑑀𑐖𑐎𑐟𑐵 ): 𑐡𑐎𑑂𑐰𑑄 𑐮𑑂𑐫𑐵𑑅 𑐏𑐵𑐫𑐾 𑐖𑑂𑐫𑐹𑐐𑐸 𑐩𑐸𑐣𑐵𑐖𑑂𑐫𑐵𑐫𑐵 𑐣𑐶𑑄𑐟𑐶𑑄 $\{E_{k}\}_{k=1}^{\infty }$ Σ 𑐫𑑂 𑐖𑑀𑐜𑐷𑐎𑐠𑑄 𑐀𑐳𑐩𑑂𑐧𑐡𑑂𑐢 𑐳𑐾𑐚𑐫𑐵, $$\mu {\left(\bigcup _{k=1}^{\infty }E_{k}\right)}=\sum _{k=1}^{\infty }\mu (E_{k}).$$

𑐩𑑂𑐴𑑂𑐰 𑐣𑐶𑑄 𑐕𑐐𑐹 𑐳𑐾𑐚 $E$ 𑐡𑐸𑐣𑐾 𑐳𑐷𑐩𑐶𑐟 𑐡𑐵𑐥𑐹 𑐡𑐸𑐳𑐵, 𑐁𑐰𑐱𑑂𑐫𑐎𑐟𑐵 $\mu (\varnothing )=0$ 𑐮𑑂𑐫𑐵𑑅 𑐏𑐵𑐫𑐾 𑐖𑑂𑐫𑐹𑐐𑐸 𑐊𑐜𑐶𑐚𑐶𑐨𑐶𑐚𑐶𑐫𑐵 𑐴𑐸𑐣𑐶𑑄 𑐠𑑅𑐠𑐩𑑂𑐴𑑄 𑐥𑐹𑐰𑐣𑐷 : $$\mu (E)=\mu (E\cup \varnothing )=\mu (E)+\mu (\varnothing ),$$ 𑐀𑐮𑐾 𑐄𑐎𑐶𑑄 $\mu (\varnothing )=0.$

𑐫𑐡𑐶 𑐐𑐿𑐬-𑐆𑐞𑐵𑐟𑑂𑐩𑐎𑐟𑐵𑐫𑐵 𑐀𑐰𑐳𑑂𑐠𑐵𑐫𑐵𑐟 𑐟𑑂𑐰𑑅𑐟𑐮 𑐢𑐵𑑅𑐳𑐵, 𑐀𑐮𑐾 $\mu$ 𑐀𑐥𑑂𑐰𑑅 𑐳𑑂𑐰𑐫𑐵 𑐀𑐥𑑂𑐰𑑅$\pm \infty ,$𑐡𑐸𑐣𑐾𑐫𑐵 𑐕𑐐𑐹 𑐮𑑂𑐫𑐵 𑐎𑐵𑑅𑐳𑐵, 𑐀𑐮𑐾 $\mu$ 𑐫𑐵𑐟 𑐳𑐵𑐂𑐣𑑂𑐡 𑐩𑐾𑐖𑐬 𑐰𑐵 𑐔𑐶𑑄 𑐮𑑂𑐫𑐵𑐫𑐾 𑐢𑐵𑐂 ।

𑐖𑑀𑐜𑐷 $(X,\Sigma )$ 𑐫𑐵𑐟 𑐡𑐵𑐫𑐾 𑐖𑑂𑐫𑐹𑐐𑐸 𑐠𑐵𑐫𑑂 𑐢𑐵𑐂, 𑐀𑐮𑐾 $\Sigma$𑐫𑐵 𑐡𑐸𑐖𑑅𑐟𑐫𑑂𑐟 𑐡𑐵𑐫𑐾𑐖𑑂𑐫𑐹𑐐𑐸 𑐳𑐾𑐚 𑐢𑐵𑐂।

𑐕𑐐𑐹 𑐳𑑂𑐰𑑄𑐐𑐹𑐐𑐸 $(X,\Sigma ,\mu )$𑐫𑐵𑐟 𑐡𑐵𑐫𑐾 𑐠𑐵𑐫𑑂 𑐢𑐵𑐂 । 𑐳𑐩𑑂𑐨𑐵𑐰𑑂𑐫𑐟𑐵 𑐡𑐵𑐫𑐾 𑐢𑐫𑐵𑐐𑐸 𑐩𑐸𑐎𑑂𑐎𑑄 𑐡𑐵𑐫𑐾 𑐮𑑂𑐫𑐵 𑑑 𑐡𑐸𑐐𑐸 𑐩𑐵𑐥𑐣 𑐏𑑅 – 𑐰 𑐏𑑅, $\mu (X)=1.$ 𑐳𑐩𑑂𑐨𑐵𑐰𑑂𑐫𑐟𑐵 𑐠𑐵𑐫𑑂 𑐢𑐫𑐵𑐐𑐸 𑐳𑐩𑑂𑐨𑐵𑐰𑑂𑐫𑐟𑐵 𑐡𑐵𑐫𑐾 𑐡𑐸𑐐𑐸 𑐡𑐵𑐫𑐾𑐖𑑂𑐫𑐵 𑐠𑐵𑐫𑑂 𑐏𑑅।

𑐳𑑂𑐰𑐫𑐵𑐡𑐶𑐳𑑃

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